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O P E R A 

GEOMETRICA 

EVANGELISTiE. 

TORRICELLII 

I>iSoUdis S^dtrMtis DifeUJoHyper^oiia 

De Motu . Cum Apfendicihus de Cy» 

De Dimenfont TnrM^ eloiae, ^ Cocbles . 



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DE SPHiERA 

EtSoIidisSphaeralibus i 

LIBKl DVO 

inquibus ArchimedisDoarina de 

Sphaera&cylindrodenuocom- 

poniair, latjiis promouetur , 

£t <» omtii ffecie filidmrHi» ,qiuevilciru,yetintr* 

Sph*Tamiex comer^onepolipm&rum re^ularium 

gignifoffltit, •miiierftlms Fro^agitur, 

AD SERENISSIMVM 

FERDINANDVM 11 

Magnum Duceim Etruriae . 

A V C T O R E - 
EVAnGELISTA TORRICELLtO 
tiufdem Serenifsimi Magii Ducis 
Mtthemttico . 



Florcnt?gTyt>« Amatoris Maffg & Liurentj JpLandisitf44 



sr tERioRv u pERM issr. 



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DE SPH^RA 

EtSolidis Sphaeralibus 

LISKl DVO 

inquibusArchimedisDodritia de 

Sphaera & cylindro denuo com- 

ponitur, latjiis promouetur , 

Et m omm ffecie filidorum ,quxvtlciru,vetmtr* 

Spb*ramiex comerfottepoliponorum regularium 

gigmpffiititoiuerfiitius Profngitur. 

AD SERENISSIMVM 

FERDINANDVM II 

Magnum Ducem Etruriae . 

A V C T O R E - 
EyANGELrSTA TORRICELLtO 
eiufidem Seremfisimi Mngni Ducis 
Mathemdttco. 



Florenf;gTy^>is Amaroris Manif & Liurent'i ip Landis i ^44 



srpBRiosyit pExuissf. 



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Sertmfsimo M*gtM Dmt EtrurU 

FERDINANDO II. 

1. SCEREM frtfdSi , Setemf. 

liUgieDKx, oitUtwia Utllmm 

Setemfiim* Ce^tiuimTtuixai- 

mm tnaximtjrciimetliii et Oa 

«eminay^iefntuaJeteff^ tu. 

c mex : Exipua emmjimt opu/ca- 

.» v»c ^ &* ^r rebusMate mHra negle. 

' Sis, tiemfe Ceometricis , Attamen , mjipillor, duo maxim» 

Ceometria operapromoueiunt, cum 'veterem De Sphtera,et 

CylindrOi nouamque De Motujiiintiam exequantur . Sed 

egofrttfiraGeometriamexcuJoapudeumPrimipem, euinon 

foiumhxreditariatfedetiamin^nita ejl Mathematicarum 

diJcipl/aariieHjgoteeiiji., iSerent/fmus enim Cofmus U. Pa. 

ter Tieusfiipendijs ceuherrimo CaUlea oliiatis; deindeSer.C, 

Tn^JtVieStifsmaximtinhuiufmodifcientia cultares colto- 

eatii,oftime dei>ionfirittiitiirteUigere,^anti momentijiat 

, ' '" A 1 Mathe- 



E4 

Mafmmatick/cieiHU iftfiiip di/paftendij atercitmm acie^ 

lmSi*V9lin mumenMsi exornandif^ nfriiSttSt vtroMte 

tempere beUi ^pacijque, Cttm enim (*vt de Mecha^ca/kcttlm 

tatefileam) totumpenecimle commerciumponderejoumert^ 

.C^ menfaraadminifiretur , quis nm yideat omne~js hominS 

negotium in Hatbematicis efie # qud tria quimtitatis^ generd 

ctminScholisnofirisquotidie agitentufj, iUiprofe^p mttxU 

me ytiUs Reip. hahehuntur, quiin huiu/modi Budtfs 'ver/i* 

ti , exercitatiqite erunt, JJhelUrum itaqtte nen ^i^iUum 

cat^a fenitus mala non eritqttatenus Geometrici/ttnt . . ifH» 

ndm mala nonfit eo nomine quod/imt mei : ProftereabumUL 

th oro ) nn Ulos quaUJcumqifint^ Tihi tamen dehitos^ r#M« 

eue munifkentia editos^ S. C* Ttta/it/cipere dignetter eo vttlm 

tUyquome quoqtte/uppUcemfitfiepit ,atque ed hummkau^ 

iqute cumtanti Princtpis maieffate coniunSla, amorem eU» 

-citetiamahignotis* FatteatDeusomnihttsyotisTtUsi C^ 

S.C^Tttam^imperiumqiditttueattn^f^augeat* 

Strenifi,C*Tute 



• • 4 



- -»lk. 



Hamililmiis fettiui 

SuauffUfia TerrieeUkul 

PROK 



PROEMIFM. 

iNTER oinniaopefaadMathematicasdirct-' 
plinos pemnentia , iure optimo Principem fibi 
iocumvindicarevidentur Archimedis inuen- 
tsiquaequidem ipfo fubdlitads miraculo ter- 
. rentanimos. VerAmimeromnesIibrosegre* 
gij Awfaonslong^cminetiUe^quiDeSphaer;^ 
& cylindro infcmMtur : nequeenim pofteritatis tantiim cooi^ 
fu , fed etiam ipfios Scriptoris iudicio pnnats tenet . Cert^ hunc 
ipie intitulumfep»rfcrielegk,dignumqj prf c^cris iudicauir, 

Siutantivirioinialumexornaretjoftenderctque. Hunctamen 
quis attentii^ conliderare, &perpendere veiir» magninn qui. 
dem inuentum feteatur necefle eft , fed fortaiTe non ^folurum. 
Loquorequidemdeptimotanciknlibro, inquopartemoperis 
Theorematicam,&omnemdodriax iauendonem cxequitun 
quo velud iai^o fundamento , in fecunda parte poftea , quaix 
coronidislocQ, problcmata qu^am tamquam corollaria ad 
eam rem ipet^antia ipfe fubne^t . Titulus libri eft De Si^uefa» 
& Cy lindro » qu; quidem verba apudnos idem fonant,ac H di-« 
xKTet De Sphata, atque vnico lolido /ph^ali ; fed fphaera^ 
Ua folida > quoium vnum eft cylindrus, multitudine funt infini- 
tX» vt mox pacebit . Ergo abfoluuor fbrtaile coniemplado vidc' 
ri pocuiOec , fi eximius Author proportioifem, non tanciim eam, 
guxeftinceriphxram, vaicumqueex fphcralibus folidis per« 
quififfecvenimetiamomnemaiiamradonem, quxinterfph^ 
ram ipfam, & vnumquodq; exinfinicis fi^uaalibas folidis inter 
<edic, oilendendfuniibiafluaipfifiet. Hocitaquepropoiinia , 
" ' erit, . 



&infUmtummeuminpnefentilibello. Dodrinamnonfolum 
dcSph^ra^&cylindro^ lcdde fphaera, & fphaeralibus folidis 
omnibus profcquemur: Mutatifq; plerumquc Archimedafis fun 
damenris , vniucrfaliori demonftrarione iDam complcdi cona-f 
binmr» atque in omni fpccie folidorum» vel intrds vcl circa fph^ 
^^^.dcfcnptorum, propagabi^ 

' Exlibro ArchimcdisDeSphaara&Cylindro duo hseccol- 
ligunturfpedantia ad illafoilda, qu« nos fphaeralia appella-- 
mus : Primum , quod fphaera dupla eft infcripri (ibi rombi folidi 
«quilateri i quod quidem vnum eft ex folidis fphaeraiibus» geni 
tum ex reuolurione quadrari infcripri»& circa diagonalem coA- 
uerfi • Altenim; quod cy liiidrus ad inicriptam fibi fphaeram cft 
fefquialter. quodquidem &vnum ex folidis fphan*alibus eft » 
genitum cx conuenione quadrari eircumfcripri , & circa ipfius 
catetumreuoluti* Stanribushsstcontemplarionedignum wi^ 
iiividebaturvnitrrfiliusaliqKiod problema huiufmodi. 

DatopoUgpno quonnque nguhrifiuemtraj^fiuleirca eircm 
lum dejiripto $ &Jiu€ tiroa dtagona^tm^fiuictrca catetumrt* 
ifoluto;profortionem dicercy quamjaffum expofygonofolidum 
habeat # adfaifam ex circulo fpbaram • 

Penitus autem cx voto iucccifitinftittitacontempkirio • Naco 
inuenta proporrione > fex ifta infcrius adnotata Thcorcmata itx 
fe habere comiperi > quemadmodum hic fubijciuntur • 

* 

PrimafolidorumfpbaraUumfpecies . 
r Siintracirculum delcriptum fuerit 
poligonum regulare habcns latera 
Bumeropaxid, &conuertaii]r figura 
circa catetum B. Qiua^itur rario ipha? 
KT ad fa^um folidom • 

Continueturrarioradij poligoni 
ad catetum ciufdem , nempe A ad fi ^"^.Jij^ A KC P 
inquamor terminis Ai B, C, D. Erit -^^ 

rtbevp^ <^^ fphaera ad fblidmn infcriptum,vt diamc- 
14J* i) icrfpharra^hoceft vtdwpla ifwus A^ad viramqj fimuIR & D, 

'^' "" SecuM 









ABC]> 



Si intra circulum defcriptum liieric poligcj- 
num regulare habens latera numero paria>& 
conuertatur figura circa <UagonaIem. A.^ 
QHaeriturratio7ph^9ad£idumfphaa:aIe fo- 
lidum. 

Odenditur • Sphaeram elTe ad folidum» vt 
quadratum A B. ad quadramm cateti A C 

Tiftia Jpecies . 

Siintra circulum defcribatur 
poligonumreguiarehabens late- 
ra numero imparia,&conuertatur 
figuracircacatemm B. Quanrimr 
ratio fph?r?*ad fa6lum fph^raio 
folidum» 

Continuetur ratio radij A. ad 
catetum B. in quamor terminis 

A, B, C, D. Eritq; fpb^a ad folid um, vtquadniplaipfius A. ad 
B.feme],Cbis,&D.femelfimulq;fumptas. *" rtw? 

Sluarta fpeeies. 'xjk. Ui^ 

Si circa circuium defcnfeatur poligonuni ^i* 

regulare,habenslateranumeroparia, & 

conuertamr figura circa catemm C QiipOT- 
tur ratio folidi ad fphaeram . 

Oftenditiir folidum eflfe ad infcriptam fi- 
bifphasram, vtduo fimul quadrata , quoru 
vnum fit ex radio D. alterum ex cateto Q . ^ 

ad duplum quadrati C xUf ^* 

^fita Species. $(. "^ 

Si circ^ circuliun defcribatur poligonum 
regulcre ha bens latera numero pariaj & con- 
uertatur figura circa diagonalem A. Quaori- 
tur ratio folidi ad fphapram • 

Oft ?;iditur folidum ad infcripta fibi fph«* 

ram 








f 

ram efle vtradius A adcatetumfi.hoc eft 



ste»t.f. fpbaere 



i 




' Sittfdacirculuni defcrlbaturpo ' 
_'■ ..ligonumregularehabens lateranu 
.'^. '.ineroimparia,&conuenatur figu- 
la circaT& catetum. QnaEiiturra- 
tio folidi ad Sphaetam • ■. 

Continueturratioradf;Aadca- ^. .^ ^ 
tctumppligoniB,intribustermints "* ^ " ^ ~ A. B ^ 
i^BiC. Eritquefplidumadfphx- 
7httr. r^n^vtAfemeUBbiSj&Cfemel fimulque fumptae, adqua- 

'tiviij.iib ^plamipnusC. 

»^ ^St>liduruhiitaq:fph?ralium fpecies omnino fex emergunt, 

«^. & vniufcuiufq; fpeciei ratio ad f uamTphaeram innotefcit . Pof- 
(eat fortafle videri tres tanmm folidorum fpecies , fi folida ab- 
folute , ac fine fuis fphjcris confiderenmr . VerUm fi illa ad fpha 
ram re^ranmr , ftatim relatio variawr , & proportio alia cour 
, furgit, iproutcognata folidis ipfis fpfiaera infcripta fiierit, vel 
circumfcnpta. 

Qmbus demonftratis, varia pro Corollarijs Theorcmata 
ftafim emergebjjnt j cm"ufmodi funt . Datis ex predi(5larum fex 
fpecierumfoUdis duobus quibufcunque , alterius ad alterum 
rationcm notam facere . 

Conum aequilatetum circa fphairam defcriptum , effe ad ip- 

.V, V famfph^mvt9.ad4. Nempcduplumfefquiquarmm. Pro- 

.^- \ •• pterea fi circa eandem fph?ram.conus, cylindruJTq; a^quilateri 

^; defcriptifint,triafoIida;nempe conum, cylindrum, & fphj- 

ram forcihter fe in.continua proportione f efquialtera . 

Sph^am ad<:dnum ^quilaterum fibi infaipmm effevtja. 
ad9. 

adinfcriptum cylindrum fquilatenHn ine^ikm 

ad } late- 




"• 4 



ns 



riseiufdem. 

- RombumfoUdum xquihcemm fyhxri^ cirannfaiptum ad 
eandemfphaa-amincomenfurabUemefle, nempc vt dumetCT 
quadrati alicuiiu ad latus dufdem . 

Spbxrale folidum exagoaale circa catetum reuolutum. efl<^ 
ad infcriptam fibi fphxrom iefquifextura . 

Sphfram auttm ad ezagonale folidum fibi infcriptum,& cir- 
ca diagonalem reuolutum , efle fefquitertiam . 

£t alia huiufmbdi, qu^ quidem alcius per&rutanti innumeni 
patebunt. Interimfatisfuperque mihieru aliqua appofuilfe» 
qua; propria claritate vltird fe fe ofFerum etiara alpema nti . Ho- 
rummaxim^pars Corollaria efle poterant pracedentium fex 
Theorematum ; attamen illa demonilrabimus ex fola etiam Eu 
clidis do<arina>fine ope illorum qu{ de iphaeralibus prsemifera- 
mus ; Vt videre eft ad Propofitiones 3 o. & 9. fcqq. in iecunda 
libro. CaeterumhuiuscontepTpIationisoccafionem,moxetia 
&fcriptionisincitaraeotumpra:buitTnihi acutiflimus librorum 
Archimcdis fcrurator Antonius Nardus Aretinus:huic enim re- 
iero,atqueipfiuseruditiscoUoquijs, fiquidvere Gcomecricij 
in hac fcriptura exciderit raihi. 

Sveropleraque mala erunt, &fortafre orania, hoc vnum 
culpandus erit ornatiflimus vir , & generc, do^ina , raoribufq; 
confpicuusAndreasArrighetws Florentinus, quipoftmagna 
inmecollata beneficia, editionem mali libri non iuaficled 
iusiit. , . __ 



DEFI- 



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Viu(cunquepoligoni<iegularis latera hdKntis oumero 
j^mayDiagonalim vocolineam, quasper oppoika^ fi* 
gi)ra? anguios ducitur « Catttum vero voco lineam , quat pun- 
aamedia latenim oppofitorum conne&itifiuc earumdem ^mif 
its. Cumfcunque vero poiigoniregularislaterahabeiitisnii^ 
mero imparia y cafetum vocolineam » cfm ab vno anguio per 
centrum figurse extenditur • 

2. Sipoiigonumquodcunquereguiare conuereatur^ fiu^cir 
ca diagonaiem, (iue circa cacetumi donec ad eum 1 oc um r edeat 
vnd e ca^pit moueri , folidum illud quod ex reuolutione circum- 
{cnbitury^bdralejotidum appellare vi&meft • Parilaterum qui« 
demfipoiigonumliabtterit latera numero paria > Impariiatc- 
rum vero, auando poiigonumlatera numero imparia nabebit J 

Si cylindniS) fiueconus» vel etiam coni fiiiftum plano per 
axem dudlo fedum fit : commmiem fecantis plani > & curuas fu« 
perfrciei fedtionem yocabimus latus cylindri» fiue coni, fiue fra 
iliconici.. 

SupfojStioues • 
. Si^{K>nimus.cuiufcunqueprifmatiscircacylindruma9queal 
tum defcripti , fuperficiem maiorem effe cylindri ipfius fuperfi- 
cie • Cylindricam vero fuperficiem maiorem cise fuperficie 
prifmatis infcripti» bafim habentis reguiarem . exceptis femper 
oafibus • Item pyramidis circa conum defcript^ fuperficiem ma 
iorem effe ipfius coni f uperficie; Infcripta? vero pyramidis & re 
gularem bafim habentis» fupponimus fuperficiem minorem ef-* 
le conica fuperficie • 

Demonfbanoir haK apud Archimedempropof. 9. i o. 1 1. 1 1 
lib. I .de Sph. & Cyl. Si quis Ver6 ea tamquam nota admittet e 
veUt, totumlibdlumnoflrumpercurrere poterit • 



DE 



". 



DE SOLIDIS 

SPHAERALIBVS 

LIBERPRIMVS. 



PKOPOSITIO F%IMAi 

I 'CyKndrkefti fuperficies fecetur (rfano op* 
pofitijjfaaiibus parallelo j eranr fegmenta lu- ■- 
p6'&ie3<!yiindric«interfe, vtfegmenta axi^ 
fineJateris cy lindri > homologe imnpta . 

- - EftocyUndnisreiaus A6CD,fe- 

ceturqjplano E F oppofitis bafibus G f''^ ' 
paralIeIojDicocyUndricamfuperficicmAEF0,ad I 
-cylindricam EBCF , cfse vt axis ad axem , fiue vt la- jj 
tusAE,adIamsEB. ^ A *, ^ 

Producatur vtri mquc in infinitum cy lindrus,& ac- "'"p 
cipiaturre^EGmultiplex ipfiusEA,iuxtd qnam- ^ 
liDetmultipiicitatcni, feftaque EG in partes ipfi E A 
SRjaaIcs,aganturperpunaadiuifionumH,T,G;pIa- ^ 
nabppofitis bafibus parallcla. Eritque tam multi- 
plex reaa GE ipfius EA: quam multiplcx eft cylmdri- 
ca fupcrffcics E L . fuperifciei E D . ^ 

Sumatur ctiam rc^ E'M makiplcx ipfiof £B>iux- 



I % De Sfhdra] (f foHdis Jphdralik 

taqu?imlibctinuidplica|:16nem j fimiliq. peraifta conftfudionc 
vtiupraj crittammultipres: redaEMrc^ EB,quam multi- 
plexeft pyHndricafupefficics EN , fiiperficiei E C, . 

• Manifeftum ergo cft.quod fi rtfCta E G maior fiief i^fiiie liii- 
nbr, vel jEqualis, rcft? E M ; tunc ctiam cylindrica fiipcrficies 
E L , maior erit , fiue minof, ver ?qualis fuperficiei E N ': & hoc 
femper : Propterea erit, vt AE ad E B, ita liipef ficies AEFD,ad 
fuperficiemEBCF.Qup^CTatd^monftwndu^ :. 

• r ; ; Ffrofppio 11 

SI fuerkquodcupqu^p^ifn[i4fe(aum,habensb^^ 
namregul&rem;habei^ue altkudinem «qualem quartae 
particatetifugbafisjeritpcriraefier prifmatisajqualispoligono 

fuaebafis. V^ ^ r 

Efto p<iligonun[i r«gukire4BCD^F, fuper quo concipiatur 
prifina redumi haBchS prt) altifcudirte AL quartam partem catc- 
tilH. Dicoperimetrumprif- 
matis, conftantem ex figuris Xi 

re^nguiis aequalibus quaru 
vna fit L B, aequaiem efse poli 
gono fuae bafis • ^^ 

Ducannir enim diagona les 
AOD, BOE,& ereda perpen- 
diculari I M , iungantur A M , 

BM; 
CumcTgoIHponaturquadrupIaipfiusIM^eritlOdHpUip 

fius IM ; & idco triangulum AGB duplum trianguli AMB can- 

dembafim habentis ; fed ctiam refianguhim LBdupIum eft, 

trianguli AMB; proptercare<aanguIum LB «qualemt triangu-« 

lo AOB ; & ficdc reJiquis rcatangulis , reliquifque niangulis : 

Qu^c totus prifmatis perimeter ,xonftan$ ex figuris rc (aangu- 

lis , asqiulis cft poligono fuae bafis • Quod crat denKMiftranau. 

CwManum. 

Co9ifia$ crgo i fu^d^aitituJofnfmaM mmmmruf^. 

fit9 





lihr Vrifttus i 13. 

fittquautquartapars catetifua bafisteritptfimetefprifmatii 
maior , minorue quampoligonumfua hafis . 

Trofofitio IIL 

SI &eritcyIindnisrectus,aiiusaltitudo ^ualis fitquartae 
parti diametti iii^I>aiis > erit cylindrica fuperfides ^qualis 
circulo fux bafis . 

Efto cylindrus rectus , cuius ba- 
(is circuius circa diametrum A B C 
defcriptus j altitudo ver^ A C, ?- 
qualis fit quattf pam dianjetri 
AB. 

Dico cylindricam f upcrficicm 
^pialem tk^ circulo fua; bafis 
AR ' 

? enim Mualis non eft ; erit circulus vel maior > uel minor cy- 
Iindricafuperficie. 

Sit primiim circulus maior quam cylindri fuperficies ; & fup 
pofita differentia G, defcribamr intra circulum aliquod poligo 
num ADEB. quod quidem. dcficiat arifculo minori def^, 
quamfitfpauumGj &ideo eritpoligonum infcriptum adhuc 
maius quam cylindrica fuperfides ( quomodo fiat hoc conftat; 
ex Commcntarijs in Archimedem , & ex XII. 'B\cM6is : ) Tum ' 
luprapoligonum ADEB condpiatur prifma recium eiufdcm 
cum cylindro altitudinis. 

Cfim erg6 alritudoprifmatis eadem fit ac cylindri.nempe quar 
tapjffsrect» AB, erit alritudo prifmatis maior quam quarta 
parstcateofu^bafispoligonaf,&ideoperimeter|«fmat!s ma* £"^f •* 
J?»7»:«q?^mpQligonumfu?bafis,&muItomaior, quam ey-^cJd. 
Imdnca faperfi€ies<&ctum enim eft poligoniim maius cylin- 
dricafupcrficiej.) iQiipdcftabfijrdHmreftenim contra pre^ 
miisasfuppofirion^s. , . . t 

Ponarur deindc circulus minor quam cylinrdica fuperficies: • 
^*W>^lrt#«^ttiaiG^ dcfcribaturiircjidituliim aliquod 
. .> " poU- 




14^ DeSfb£ta,&/oIfdisfi^^aM 

poligohum regnlare D E F ^stl-^' ^^ 

C^uod cxcedat circulmn ''^ 
fpatio minori qudm fit C . 

(quomodohocfiat conAat 
apud Commentarios in Ar 
chim.&in XII» Euclidis.) 
eritq; etiam poligonum mi- 
nus quam cyJindirica fuper- 
ficies* 

Concipiaturfuprapoligonumerigiprifmaciafaemaltrtudt- 
mjoimcylmdrojeritque altinidoprifinatis quatta pat» catetE 
lu*bafispoIigon?.(cuin prifmatisaltitudoeadem foatq: cv- 
imdrii cylindn autcm altitudo eft quarta parsreca» A B. 

perii.bu ^"?f^"^^<^^^<>,P<>%on^uodeflbafi$prifmatis.) 

^ ; Ideopenmeterpnfmatis «jualis erkpoligono fue bafis j & 

proptereaminorqudmcylindricafuperficies. Qupd eftcon- 
trapraemtfrasfuppofitiones. 

^Erit erg6 fuperficies cylindrica asqualis citx:uIo fuc bafis» 
Qioderatdcmonftrandum. 

Propo/tfh IV. 

GYlindriredifupcrficies ad circulumfu? bafiseftvt latus 
cylmdn ad quartam partem diametri ciufdcm bafis . 

Efto cylindrus redus , cm*us re<aanxnilum a 

" lar ^ 

fiaem ABCDadcircuIumfua; baS cfl^TvL.' 
ABadBE. ^ 

' ProduaturcyIindnisvcrfusF,fe<aaque ^i 

BF»quaIiipfi]BE,critpcrpntt:edcnten3y- P 
hndrica fuperficies F C aqualis circulo fua 
fetihth^ bafisBC. T . H* 

^. ' ^«>-«ylindricafupafid^BP^cyl^ 

FCeft 




LiherVrifMU' \% 

F C cft>t ABadBF jfuperficies vero FCadcircuIumBCCob 
(qualitatem) eft vt F Bad B Es Ergo ex^uo erit cylindrica fu-- 
perficies BD ad circiilum B Q vt AB ad BE, nempe vt latus cy-* 
Indriad ^dianietribafiseiiUdem.Quoderatoftendendumi 

Frofoptio V. 

CYlindri redti fuperficies ad circulum quemlibet » eft vt xt^ 
dangulum per axem cjrlindriad quadratum £emidiame- 
triipfiuscirculi» 

Efto cylindrus redlus cuhis xt(^XRr^ 
gulum per axem fit Al^ & centnlm ba ^ 
fisH. Ponatur auftm circulus quili^ 
bet cuius femidiameter CD « Dico cy 
lindricamfuperficiemad circulumex f 
CD» eflevtreiftaagulum ABadqoa-- 
drawmCD* -A. EH 

FiatexAE (quasr quidem 4«pars 
fit re(S^ AL ) quadraajm F£» pix>diKaturmie £G « 

£rit ergo cylindricafuperncies A Bad circulum fu^ bafis» vt fn tr^i 
lAadAE»hoceftvtIAadAF»hoceftvtrc<aangulumJIEad *'^- 
quadratumF£tfiue»ium^quadruplis> vt redbngulumAB rrim^e^ 
ad quadratum ex AH « Cu^cttlus verd bafis A L ad circuhtm cy 
CD r eft vt quadratum ex AH ad quadratum ex CD : ergd ex %.du^ 

fsequo erit cy tindrica fuperficies ad circUlum ex CD > vt redan- «« • 
uIumperaxemadquadratumCD» C^pd erat demonftralK 
um« , , 

PttCof0liariQtritPt9Jf$fiMXI/mk i^Archim* deSphs^ 
tm^ Cytkndtik « Cmftat tnim qnidfiCD^ mtdiafutrit tpropor^ 
tionalisinttrlAy AL; qnadratum tx COaqttak trit rtffangn 
U]AB. iyprofttrta^txdtmQnfiram^tj^dritamJuftrfiQkm 
AtHl Sfmkm tfi sirnUbtx[C7> nrttfi tjf. 

» 




Frodifttio VL 

CYIindrorum ruperficies inter fe funt vt eoruindem redan- 
gula per axem homologe fumpta . 
Smt cylindri re<^i quorum re^ngulaper axem fint AB,CD . 
Dico cylindricam fuperficiem AB , ad cylindrieam CD eflWvt 
redtangutum AB ad rei^angulum CD . 

Accipiatur pro circulo quolibe^ cir o 

culuscircadiametrum AE — *" ■ 



~\\) 



E it ergd cylindrica fuperficies A B 
«* ad circulum quemlibet AE , ut redbng. 

ABadquadratum AF.CircuIusvero ex _ 

AFadcylindricam fuperficiem CDeft A- F E C 
fer frd* Vt ^uadratum ex AF ad rei^gulum. 
•"'• CDjergo ex^uo cylindrica fuperficics AB ad cylindricam 

CD.eftvtredangulumABadredang.Cp. Quoderatoften 

dendum. 

TrofofitfoVIL 

SI redaf^amisbafimhabueritpoligonamregularemque 
erit bafis pyramidis ad reliquam ipfius fuperficien^ vt femi 
catetus bafis ad catetuin fuperficiei . 

Efto p)Tamis re&aiCuius ba- 
fis poligonum regulare AFED. 
vertex verd G, & centrum bafis 
fitl. St&o deindc vno laterc 
bij&riam in HjiuniSifq; GH,IH, 
erit GH catetus fupernciei pyra 
midisJH vero femicatetus bafis; ^ - » 

quandoquidem omnia ttiangu- 

la in fuperficie funt acquicniria» & sequaliaiotei: ki quod etiain 
vcrumeft&inbafi. 

Dico. 



Dko bafim ad fupcrficiem cfle vt I H ad HG - 
Triangulum coim A I F,ad triangulum A G F ( cum fint in ea 
dembali)eftvtIH,adHG, ergo etiam ipfomm fquemulti- »,. 
plicia,ndmpeb'afis»&fuperficiespyramidi5,ineadcm-rationc "• 
cnint>ncmpevcIHadHG. Qw>d crat oftcndendum. 

Trofoptlo VIIL 

COni oe^Ti bafis adxcliquamcbnicam fuperficicm, eft vtfe- 
midiaioeier bafisad latus coni . 

Efto conus re^ , ciuus ba- 

fis AB,veriex vero Caxis CD. 

Dico circulum bafis, ad rcli- 

quamconicam fuperficieni,efle 

viDA,adAC- 

Si cnim iia non eftjcritcircu 
lus Afi vel maior,veImin.qua 
oponete0c , vtadoonicamfi^ 
pcrficiem fir CjUcmadmodumDAad AC . 

Sitprimummaiorj&ponatiffCamomaiorquantum eft/pa- 
cium H. Infcribdtur in circulo poUeonumdefu:icns d circuTo j 
minoi:idefe<auqQamfpacium£;h^ebitq;huiufmodi poligo- . 
numadcoflicam fuperficiem adhucmaiorem rationem, quam 
D A ad AC« Se^ dcinde vno poligoni latere AF bi&riam ia 
HiiungannirDH, CHi&fuperpoligonoconc4>iamrpyramis 
qux vctticcm babcatin C; feceturque DI ^ualis ipfi DH,& du 
catur I L paraJclJa ad BC, imiganirq. IC . 

Cum itaq. poligonum ad cOnican? fiqierficiem maiorcm ha- 
beatrationemquam D Aad AC; multo maiorcmrationcm ha- 
bebit ad fupcriiciem ll^ pyramidis,quamD A ad AC , vel D B 
ad BC> Sed poHgonum ad fuperdciem pyramidis , perprxce- 
dentera, cft vt DH ad HC jhabebit crgo DH ad HC , fiue D I 
ad IC , multo maiorem rationcm 4juam D B ad BC> vcl quam 
DladlL. £tpfOptereaICmiaorcfiecquknIL. abfurdup^. 

C Nam 




rt ZV Sphdrd, (ffiliJiifih/tralii. 

Nam quadratumlC asquale cftduobus quadratis ID,DC! 
cumquadrammlLieqniilefittintam duobusID.DL. Pona- 
tur deinde ci rculus bafis AB minor quim oportct efle vt >d co- 
nic am f uperfici em fit qucmadmodum refla D A ad AC , filque 
tantd minorquantum cft rpatium 
E. CircumfcribaturcirculoAB 
poligonum aliquod cxcedens 
circulum minori excefsu quam 
fitfpatiumE. Habcbitq.poligo 
num ad conicam fuperficie,ad- 
huc minorem rationcm quam 
■ DA ad AC; cr^6 poligonum ad 
perimetru [ax pyramidis mult6 

mioorem rationem habebit quam D Aad AC. Sed poligo- 

rtri.tm nupiadperimetrumfiljpy^amidiseftvtDFadFCi proptcrca 

""• DFaiFGmuItominorSraiionemhabebitquam DA ad ACi 

quodeftimpolTibile. Aequ.-UcsetcnifflfunttamDF, DA.in- 

tcr fc i quam^EC^AG i intcr fc . 

Erititaquc b.-ifis coni recfti ad reliquam fupetficieni, vt D A 
ftdAC. Qupderatdemonftrahdum. 
Ccntlarium* 
Hmfatct quod tututJuftrfitUi tnii^qualii tf tircult tui- 
dtHHituiuifimidiamettrmtd.prop.filittttr CA,AD, tttm- 
pttiitter lalut, ifjimidiamttrum bafis totti . Namfumpta mt- 
4iainttrCA,ulD,trittirtuluiquifitixmtdia,adtirtutum 
quifittxATl.vtCAadAb. Siditiam turua anifuplrfi- 
f„ f,,, titi,<idiirtutum tx AT}, tftvt CAadAD. Ergt nqualit 
«rf. ificurua ntiifuperStitittirculo, cuiuifimidiamtttrmidiafn- 
fiirtit»aliifiti«tirCA,AD, 

PropofitioIX. 

CViusIibetconiteaifupcrficics,ad fupcrficiem cniufcunqi 
.cylindri reai demprs bafibus , cft vt reaangulum lub b- 
terc,& fcmidiamctro bafis coni , ad reaangulum per axcm cy- 

lindri, „. 

Efto 



EfteconusABCcuiusbafisAQaxJs ** ^ 



,«- B rEi 

lumperaxemfitDE.DicocofUcam^jh /K i I 

■*• /IM I 



vero B H ; & cylindnis cuius tedangu^ 
lum oer axem ut DE • Dico cotticam &h 
per ndem ad cy lindricam eAe>vt redaa« ^ 
gulumBAl^adredan^ulum D£» i^TlHnC D 

Nam conica ii^er^es ad •circuittm 
fuigbafi5eftvtAB>adAH,iiueytre(fta fttiJm 

gulumBAHadquadraounAH. ciixrutusauiemeK.AH/adc^- '^* ^^ 
iindricam fuperficion DE, eft vt qiiadratum A^ad re^angu* ' 

lumDC Propterea,exjequo,eritCDnica fuperficies ABCad 
cyiindticamDE; vtre^angulum BAH ad re&angulum.D£« 

Quod erat oftendendum • 

■ 



mi. 




COnicifiipMidksJl^^^ vtfeftln^ 

guiafublatetibusconorum> &-fubfcmtdiaaictris hiSSb 
coiripraekeiifiu '. •«' ^— - r .r:> ' \ . i ' 

Sint duo coni re<5ti ^QD£F^.qtto a 

rum axes BG, EH • Dicocuruamconi ^^ 

ABC fuperficiemt ad*curuam fop^^ • 
ciem coni DEF ; eflfe vt ivdhn^akjm' / \ ^ f^% 

BAG,adre&angul^ / \ /f\\ 

nurumiiibiatenbas coBQrum,.&£&* / I \ X I \ 
midiametris bafium compnehendun^ A G . C DH Jf 
tur. 

Conicaenimfupicrficiej^AiBCadcircitfqfii AQeft vt re^ j^ g^ 
BA ad AG^fiue vt re^oguhirti BAG; ad cw;!^ AG « Cir* ius. " 
culasver6 ACadDFf:iteuWif^ft vtqiB«an|m AG,adDH;J 
deniquecirculusDittJidc6nic.am fuperficiiemr DEP»eft vtqua- /fr SJW 
dratu^DHt^d redangulum £B|Jt^j^g6 ex asquocuruaconifi)^ ^^ 
perficies ABCadcuniamDEI^» drft vtre^ai^umBAGt ad \ 
re^gdliimEDH* Quodei^oftendenduQi. 

C s Lem^ 



to 



De SfbdM^ & JoUdUfphxrdliB. 



^^ v# 




Limma. 

SifueritABCDfruJlum^omnai^i^ . 
fcijfum planis ad a»em ere£fts ( bec emm 
modofemper intelligemusfrufta eonica ) E^ 
fecenturque latera A B , DC bifariam in g 
fun&isEf^H.iut§gaturq\EH . Dico 
reffamEHc4fmponfexv0raqueBt^A/p 
mmpe exfimidiametris Imfistm oppofitar umfrujpe ponici . 

/ungantur^BDiE/fLHiEtquoniam ui/$ /D. £qttak$ 

f^^. ""f^funt \ item AEyEB » aqud/es: erunt para/lela £Jr BI>. iirideo 

inparal/etog^ammoaqualia eruntlatera /DtEAl . Obean^ 

dem caufam aqua/iafunt ^L^MH . Ergo tota EH aquaiis erit 

ipp /D^BLfimulfumptis^ ^wderat^/rc. 

befinitiines. 
- Vocaiimasimpofterum bfmtati§cai^ Usieam EUmedii 
tjfritnuHeamfrufiiconicr. 

Reffangulum wrofifb EH & A^BlatereJru/li^omcii dki^ 
mus redlangulum propriuoi fiufti cohici • 

Frofofttio XI. 

Cyruafuperficies fhifti conici, ptanis ad axem tr t66$ ab^ 
fdin, ad coQicam quamlibet ruperficiem,eft vt re(ftangu« 
iumproprium£^uftiyadre^an^ulumfiibiatere><'' ^* "' 

bafis^ipfius€OBi*- 

Efto fiiiftum conicii 
.. • s'*- ABGDabfciirumplinis * 
• •ad axem eredis, ^ue / /j/7 \ w 





A 



(fckm quilibet EFG,cu 
ia^axisFH. Dicocur» 
vm £taS& AC fuperfici- 

cfn^ad cuniS coni EFG ■ a t ~T\r \x ^ 

fuperficicm.cfleA^t reai vA ;, *-. »«. H O 

gdimfubA54cfubwwqucAI*BIcomentuii^ adredangu 



Gompleatur contts AMD. cuius d^tum er at fiuftum , £idoquc 
anguloMANrcda» &fedta AN cquali ipfi AL . compleatur 
redangulumAP. Dudo deinde diametro MN, & iadaBO 
parallelaad ANxrit BO aKpialis ipfi BI.con^leaiur etiam figu. 

raBQ:^ 

lam fuperficies curua coni AMD ad fuperficiem curuam co- f. **' 
ni BMCeft vtreOangnlum L AM ad re^aangulum IBM ; ncm- **""* * 
pe vt re<^n^um AP ad BQ > & diuidendo, ent cufua fiiifti co 
nici ABCD fuperficies^ad fuperficiem coni BMQ vt gnomon 

A O P,ad re^angulum BQ^hoc eft vt re Aangulum fub A B> & 
vtraque AN , BO , fiue AL , BI , ad re(5hngulum IBM . Curua 

ver6 fuperficies coni BMC ad curuam coni EFG, cft vt re<aan- 
euLIBM ad rcc^FEH.erg6 ex aequo curua fiufti conici ABCD 
wperficies ad curuam coni EFG luperficiem cft vt re^n.con« 
tentamfubAB,&vtraqueAUBIadredanguIumF£H. '^ i 

Cwollarium» ' 

Pstet crgo quodfmfii comci A^CD fupetfaicsfitu bafibut 
sifuftrficiem coni EFCtfivt rtfftmguiumpropriumjrufii ad 
fthangulum FEH. RtffangulumMttempijopriumJrufiicom» 
frebenditurfub rella ABt&fub vtraq; A LiBl,fiutpotiusfitb 
A^,ir mtdia Aritmttica, quam dtmonftraumut aquaUm v* 
trifipteAL,BJ, 

PropefifioXIL i 

CViufcunquc fruft i conici fupcriicies ad fupcrficicm cylin^ 
dri rc^ cft yt redangdum propriuih frufti ad rcdaogu^ 
lumpcraxcmcylindri* . . 

£fto fiuftum CQnicum ABCD » 
& cylindrus cuius rcdangulum pcr 
axcm fitEF. Scccnir AB Diiaria m 
H^&aganirmcdia Arimficdca HI 
{quidmanterad BC. Dico coni* 
cam fiiifti fupcrficiemf ad cylin-» 
dricam £F,clfc vt roftangulumfub 
H]«&AB^adrca3t^te£F; 

Acci^ 







sr De SfbM, tf fehSs ffh^sUk , 

Accqnatur conus quilibet LMH cuius axis MO. Eritq;cnr- 
fer frd- ualruftifi^erficiesadcanicam curuam LMN#vtredangultini 
* ^ fub AB) Hly ad redangulum MLO^fed curua cani LMN ad cur 
uam cyiindri EF fuperficiem , eft vt reftangulunfi MLO^ ad re- 
^ngulum EF^ergo ex aequo curua frufti conici fuperficies , ad 
curuam fupeifkriem cy iindri^eft vt redanguium fub AB» & HI » 
nenvpe vt re<^gulum proprium frufti, ad re^angulum £F per 
axemcylindri» Quoderatoftendendum» 

CmlUrmm. 
Curua fuperficies cuiufcunq; ^y ' vC 
frufti ^onici ABCD aequalis dcs 
mondramr circulo cuidam , cuius 
quidcm circuli femidiameter E 
mediaproportionalis fit interia- 
tusABfiiifliconici» &interFH I j 

media Aritmeticam eiufdem fiu- T | j 

fti/ — ^ 

Edo quadratumlE aquale redangulo fub BA>FH « fumatui^.^ 
que cylindrus quilibet IL ; & erit curua fiufti conici fuperficies 
fif ffs^ ad curuam cylindricamlL, vtlredangulumfub BA, FHad re- 
Tlftftf/. ^^g^^ IL ; fiue vt quadramm E ad re^flangulum IL ; hoceft 
' vtcirculusexradiofi^adcuruamcylindricamIL«Aequaleser- . 
g6 funt inter fe curua fiiperficies fhifti cpnici AQ& circui us ex 
radio Efadus . Qua^ quidem Arctiimedis Propofitio eft 1 6.. ii« 
bri primi de Spfai. & cyL 

Tropopth XllL 

S^i. rirniliim tetigerit reda qu^iajn linea «qnaiitBr vtrimq; 
^ |)roductas&conuettaturdrculus circaquemlibet fuiax6r 
( dunimodo^^axis tangeme^n non fecet ) erit conki fiikfti fiiper^ 
ficie4qu;r altangetite linea defcribitur , aqiialis fuperficiei C)r«l 
lindrl eandem alutuHinem cum frufto conico h^ntis^ & circa 
eandemfphaaraindefc^ptibilis. ^ 






cifculiis iSDBQ quem du«diatnetri AB)CDfeceAt 



angtt- 



T » -■ 




UkrTnmm. -tj 

atigttlos rcAos • Duas infuper cuigcntes habcat alteram DG 

snextrcmkate diamctriCD» alteram vero vbicunque inl>& 

(qualicer f>roducamr hinc inde IL 

IMidumodo axem AB produ- 

dum non fecent. Agantur deinde 

per L, & per M parallelae ad CD, 

re^LE,MF. tumfigura con- 

uertatur circa axem Afi • Tan- 

gensGH defaibct cylindricam 

quandam fuperficiem cuius re^ 

gulum per axem erit EFHG:Tan 

gens vero L M dcfignabit fruftu 

conic^ fuperficiei s deniq> circu- 

lus ipfe fphoeram circumicribet • 

Dico cylindricam fuperficiem a linea GH defcriptam , & com^ 

cam fuperficiem a linea LM fadam aequaies elTc inter fe • 

Ducatur IP media Aritmetica conici frufli ; & agatur IR per 
centrumQieritq; IR perpendicuiaris adLM: Ducatur etiam 
MT perpendicularis ad £G • 

QioniamduoanguliTMI ,T LM vnircifio funt asquales, 
ncmpe ipfi L IQ^demptis alternis TLM,LIS, cnmt aequaies re- 
liqui TML, SIQadeoquc triangula TMZ.> SIQ^cum re<£hn- 
;uJafint, fimilia erunt j Ergo vt T M ad M L ita S I ad IQ^ 
loc efl ( fumptis duplis ) PI adlR: & ideo red^ngulum fub 
TM ,1 R (quod quidem eft rcdangulum EFHG.) equale erit 
re^ngulo fub ML , IP . quod proprium vocamus fiiifli conici • 
Proptereaper praxedemem a^quaiis erit fuperficies conici frui* ^ 
fti , qug a iinca. ML defcribitur , fuperficiei cylindri EFHG,ea- 
<lemaltitudinemcumipfo£iiftohabentis,&circaeandc fpb^ 
ramADBC^defcriptibilis^ Qupd&c« 



Vropofitio X I V. 



* »• 



s 



I circulum tetigerit reda linea »]ualitcr vtrinq^ produ^ 
& conuntatur circuius circa aK6i%qui cum tangente con* 

ueniat 




14 7)e Sphdhra» tf filf^fP^rdiik 

ueniatin cxtremitate ipfius tangcnris » enUuperfiaes com» quc 
atangente defcribiwr , «qualis fu.crfiaei cylindn, eandem 
cumconoaltitudinemhabenas.&circacandemsphaErAm dc-. 

fcriptibilis. 

Pofitis ijfdem vt in prarcedentis 
propoficionis conftrudione ; fi li- 
nea MLincidac in axcm B L pro- 
du(5tu>fintq;aequalcsvtrinque IL, 
IM^tuncdefcvioet ipfaML coni- 
cam fuperficiem» Dico conicam 
huiufmodi fuperficiem asqualcm^ 
effe fuperficiei cylindri E F H G* 
eandem aititudine iiabentis cum 
ipfocono, &circa eandcmfphae- 
ramdeicriptibilis. 
Fiat cnim^ngulus LMT re<ftus, & cufli LM dupla ponatur ip- 
(ius LI» erit MT dupla ipfiiis IR , hoc eft «qualis diametro fphf 
ra^9{iueip(iFH.cumautcm , perquartafexti,fitvtML adLN, 
Ita TMad MN.erit redangulum LMNa?quaie r edtangiilo fub 
TM, LNt boc eftre^angulo fub FH , L N, quod quidcm per 
axcm cft cylindriEFHG • Acqualis ergo cA fiipcrJScics coni 
f^btdusi OLMffupcrficieixyiindriEFHG^ Quod&o 

Trofoptio XV. 

SI circacirciJlttmdelcribatitt-poHgonamhabenslatera no- 
' m^oparia,fiueaquatemario menfurentur^ fiue tantum 
a binario , & conuertatur figuracirca diagonalem, erit vniuer^ 
jfa fuperficies fa^ fphaeralis folidi , o^quaas fuperficiei cyiindri 
circa candem ipbasramdefcriptibilis^ 

£fto|K>ligonum ABCDEF « parilaterum ^ fiue d quatema-^ 
rio numerus laterum menfiremr , vt in piima figura,fiue tantum 
abihatio^vtinfecuadai&coauertaturfiguracircaaxem AI^ 

nem* 



lihr trifiiftil 



^i 




nempe drca 

diagonalem 

poligoni . Di « 

co vniuersa Jf^ 

fuperficiem 

fedi folidi 

fphaeraiis ^- Ct 

quolem effe 

(aperficieici > 

lyndri GH 

IL eandcm 

altimdinem 

habenmci:miprofolido9&circaeandem fphaeram defoipti^ 

bilis, - 

SuperficiesenimconiBAF»aequalis-eft fuperficiei cylindri ^^ 
ML ; Superficies autem fruft i conici , quas inter plana BF » G £ 
intetcipitur /^uali^ efl fiijperficieicylindri' inter eadem ptana 
intercepti:& nc de finguiispartibus faperficieriim , quae foli- 
duni fphanralc circumfepiunt ; Ergo omnes fimui fuperncies am 
bientes fphaerale folidum a^uales erunt fuperficiei cyisidri 
G H 1 1. • Qgod erat oftendendum • *" 







A 1^« 



Si circulMm dnd Sdmefri jiBy CD, f^M- 
g^los reiJos secueriMy umdemq; circulum 
du£ ^quAlesreCtA liued AF^ SG tetigerinf in 
txtremifdtihus dxis AB. Tumfgurd circji 
dxem AB commAfuf , iticrihtnt AP , SG 
duos fircuios dqualesj cum tfsa Aqualesfint. 
Ofortet segmentum cylindri csrca edndem 

sfhstrdm descriftibilis reperire , cuiussufer^ 

fcies dquMlisfit duohusfimulcircuUs ex AFy ^ 
BG descriftis . 

FidtdrigulusHGIrtBuSyCrhqiBldtitud^qudfiti cylindri 
Hdmfroftcfdngnlum rcShmfiGJ^ erit re^dngnlumffBJdqud 

D iequ4^ 




f4 D^ Sph^Sf tffilidkfphdrali^: 

leqM4JrM$ BG i&reifd9$gmUm ABlhoc eB reffM^MUmLH 
iJkwim: dufUim erit amddrdH BG. Profterea smperfcies cylindri L U dsu 
fU erit circidi exBG dtscrsfti , dr ideo d^itdlis lom^oktts ^ircst^ 
hsex^G^AF JimHlssmftis . ^lgeddrc 

Profofim XV L 

w 

Sl circa circtilum defcribatiir poligonum habens latera ni^ 
ioeroparia , (iue p quarernario !kieDfurentur»fit)e cantum k 
binlario » & coiuiertatiir ngora circa ciatemn» qrit vniuerfa fiiper 
ficirsiFadi^^aeralis j(cilidi. » ^qualis fuperficiei cyliodri circa 
eandemfpbseramdeicripubilis»altitudinemver6 habentis a>-^ 
qualemlinea?co<npQfitieerdiattietrofpha3r«,&ex tertia prp- 
portionalium » fi fiat vt fphazrse femidiaineter ad femilaaispoli- 
gom» tta feixiUaiitf adaHam* 



E A H 



B 



p 

H 



Y_ j[ 



JM. 




ibcemdi^dia 
diaxofttnAC^ 
BD adangul. 
redosy&circa 
ipfum fit poli^ 

fonafiguraha 
ensl^eranu 
meropana^fi^ 
oeaquacema- . ' 

Ho menfurentiTt vtinprimafigiira ifnie tantimi abinario» ¥t m 
ibcupda: Tum cbniiertfitir figuracircacacetiim AC^ hoc eft 
circa liiieam coanedentem biiedioneslateiiimoppofitonims 
£x reuolutione poligom folidiun i^^hasrale ddCcrib^ur conten* 
twn fub circuhiribus» cooieiiipie ioperfidebus» & voacylitidrl*' 
ca, vtinprimafigura,fiuecirciilaribus»&conicistamiiiD»vtia 
fecfmda. Fiatdeindevt/CadCX^]taCXadCir9<]uodfacile 

ere- 



mAom-ad axem #Dico vniuerfani iuperfidcmfolkti^^faaeralii 
«qualem efle fiiperfidei cylindri ENOH » 

Hoc aucem patet ex prasmiifis i Nam tota fpberalis folidi fii^ 
perficies » demptis drciiiis oj^fitis » asqualis eft fiqKrrfidei cy^ 
hndncx inter piana Ef?» FG con^rashenfar ^ Duo verd cwcu- 
li oppofiti quonim centra A> & C asquales funt (perpraxeden^ 
lemma)fuperficieicyiindrica(interduoplana FG»NO« con*-' 
tentf • Propterea vniuerfa fimul fphaeraiis foHdf fuperficies as 

3ualis erit iuperficid cy lindri ENOH • circa eandem ijphasram 
efaipti, & altitudinem habentis AM, quas componitur ex dia-^ 
metro fpbao-a? jiC » & ex ^eda C M , quat quidem tertia pnopor^^ 
tionalis eft ad femkiiametrum Ic; & feiniiatiis» CL . Quod &a 

S$ circttUim ASCD dH^diamttri AC , BD 
secentAd dngmhs reff^s; reifd dmtem lined 
CE eOdem centingdt in extremitdte dxis AC 
& connittatnrjiptrd etrcd AC; ifsd CE cirem 
Inm descrtbet . Ofmet segmentum cylindri 
circd edndem sfhttrdm descrifti reperitey cn^ 
inssnferfcies dqudUsfit circnlo ex C E de^' 
scrifte . 

Eidt dngnlns AEH re^ns , dnCletjnt fUn$ 
ferHdddxemereSf$. iHco cylindricam sn^ 
ferfciem MILN . acfndti circnle ex CE . FJf enim p^ dngnlnm 
teilttmAEH yreCtdngnlnmACH yhec efireSfdngntnm ML^ d^ 
qndle qnddrdte CE . Proftereifuferficies cyUndriMILN nqnd^ 
liseritcircnleexCE. ^npd&c. 










"Propofitk XV 11. 

^ « 

•r 

SI drc2circalumddaibaturf)oIigoiHmlu^]«ntIia^^ 
mero imparia, & conuertamr figura cirdi catenim poligo- 
ni : erit vniiierfa inperfides fadi fpfaaeralfs folidi cqualis fuper- 
fidiet cylincbi circii eandem ipb^ram defcriptibfli», altiaidinl 

D a verd 



49 De Sfh^.^fiMirthAralih. 

yero habetuis^tequalem Uae« compofita^ex cateto poligoiu/^ 
ex tertia proportionaJium , fi fiat y t diaoneto- circuli a4 /emila.^ 
tus ppligoni;itafemiU(us ad aliam. 





B 


^^ » 


• 


* < 

4 


V 



N 



tlM. 



/ 



^ : Eftp ci^culusX^A^ O . E 

• ^tppiigooMm Eif QHI. haben^^ 

ca ni^tieipinipari^ 5 & conycrtacur 
* figura circa catptum £Q nempe cir 
ca linea, qua? ab vno angulo £ pcr« 
ducltui; ad bife^onemlateiis oppo 
ini,^ pr ietprq; A4i<}iun fphaerale con 
tejxvoi /ub <;pniciiS: l^perficiebus , ^ 

vnicoque circulo • 0^^ a^^^L^^ — I % 

Fa^o deinde angulo redo jtffLj jyc 
dudoq;per/^plano MN ad axeni 
eredo« .Dico vniueriani fol|di fuperficiem asqualem efTe fu^ 
p^rficiei cylindri OMNP • 
^ I Nam fuperficies folidi ^baaralis, dempto circulo ex CH. dc- 

' ^' fcriptb , aequatur fuperficiei cy lindri iuter plana OP, QR con- 
tenti : circulus autcmextC H fadfais ^ualis eft (prseccdens lem- 
ma ) fiiperficiei cylindri iater piana Q^ MN cotitenti: Propte- 
reavniuerfa folidi fuperficies ^qualis erit fuperficiei cyiindri 
OMNP.qui quideii^circa eandcmi{rfH^am cum ipfo folido de^ 
fcribitur,altitudinemver6 habetlineajnfi^^qu^componitur 
cxtateto£<7v&;alineaCZ, qua^terda propordonalis eft, fi 
iiat vt AC diameter fphaeras , ad CH femUatus poligoni > ita €H 
ad aliapi f V QSiod 4erat &X:. 

Vropontio XVIIL 

Hfmisphaa*!! fuperficies ^ualis^ft fuperficieicuru^ cylui- 
dri eaad^m ipfi bafim, & eandem aititttdinem habenec 

jEftohemispfaaeriumwf^Ci&circaipfum cyHndrus eiufdem 
altijudinJ5^^i);EC, \ ^ ■ 



*9 




HA 



LihrPrimus. 

Dico fuperficiem hcmifphc- 
rif^qoaiem eflTc fuperficici cy- 
lindri ADEC^ 

Si enim non eft aequalis, vel 
m^r erit , vel minor . Pona- 
tut primihn fph^rica fuperfi- 
cies maior : fiatquc vt cy iin- 
dri fuperficies ad fuperficiem 
hcixiiipherij,quaemaior poni- 
tur,itare^ AD ad AG: intelliga- 

turq;cylindrus produ<ausvfqueadGF* Secfeturdeinde arcus 
ABbifariam , iterumq; portiones eius bifariam , & hoc femper, 
donec poli^oni circa femicirculum A B C dcfcripti femilatus 
VL minus fit quam re(fla DG . (quod fieri poffe conflat cx pri- 
ma Decimi) iemilatera enim poligonorum circulo circumicri^ 
ptorum ex condnua arcuum biie;Slione femper minuuntur pluf^ 
quam pro medietate^ vt ab alfjs oft^hfum e({.)Fa6lum ergo fit > 
& efto poligonum HILMN, conuersaque figura circa axem 
LO , fiat ex poligono, femifolidum fphxrale fub coincis fuper- 
ficiebus compra^henfum* Cum itaque red^i DG maior fit qua 
fcmilatus L V, multo maior eadem erit qutim LB , & propterea 
pbnum PQ^ produftum per L intra ^m&x D & G cadct . 

lam quia fuperficies cylindri AE ad fupcrfiiciem hemifphfE*- 
rij eft vt A D ad A G , hoc eft vtcylindrica fuperficies A E ad 
cylindricam AF,eritcyIindrica fuperficies A F g^qualis fphasri- 
cae . Propterea, fi fphaerica fuperficies a^qualis fit cylindricae AF# 
maioreritquam cylindrica AQjioccft quan^conico? omnes 
HIZ.MN,muItoq;maiorquam omnes AS I L M R C^ quod 
cii abfurdum • Eft cnim contra principium ab Archimede prae^' 
miisum. . . ^ 

Affkfnfjinms comcdm qu4t defcribitur i linea HS mdinrem ef- 
fe quhn ilUfHperfcies ,^jfie defctiiim k^^ned A S. quod fdtet 
exii. hniHs . EeifdngHlHm enim froprium conic£ fnperfciei 
. mnlth mditts eB qudm reifdt^ulHmper dxem cylindricfi qjndndd 
:ptUcmfHb mdioritm Uteri^tts continetifff ^ \. 

Pona- 



f, Mas 



ix XV, 
buius . 




Ponatur iam fphasrica ABC 

mmorouacylindricaADEC. ^q^ 

Fiac vt iupcrfTcies cylindrica-* 

ADEC adfphxricam^qus 

pooitur minor; ita rc£ia AF ad 

FL.Fiatque ex FL femidiame- 

tro aliud hemifphan-ium LNI> 

wiori conc cn t ii cam / & circa 
ipfiiin intcliigatur cylindrus 
L H M I : Infcribatur etiam 

inaafeiiiicircuIumABC.figuraIateram«qualium, itavtlate- 
ra ipfius non tanganf femicirculum L N I . (quod fieri poflc 
conftat ex Euciide • ) Defci ibaturq; aiius femtcircuius femi^ia- 
metro FO, qui contingar fingula latera fa&x fijgurg , &:conuer«- 
tatur v6iuerfa figura circa FB . ita vt fiat femifolidum fph»:ale 
AVBTCconicis fuperficiebus circumfeptum; ex femicirculo 
autemFOfrataiiudhenufphaaium»drcaquod condpiaturcy* 
lindrusRQSP, 

lam fic; fuperficies cy iindri ADEC ad fiiperficiem hcmif^ 
ph?ri j eft , per conftru Aionem, vt AF ad FL, hoc eft vt AC ad 

^ LI, hoc cft vt redhnguium AE ad re(5bngulum L M,hoc eft vt 
iMfUf* ^ijn jj.j^^ ^g ^j cy lindricam LM . Qgare fph^rica fuperficies 

fqualis erit cy lindricae L M» & propterea minor quam cylindri- 
#ft i5.fo ca R S> hoc eft quam omnes conic? AVBTC, abfurdum.fph^ 
^ • rica enim fuperncies A B C maior eft quam onmes conic? AV 

BTC, 

Hemifph^rij ergd fiiperfides foualis erit fuperficiei cylin- 

dri eandem ipfi bafim , eandemq; altitudinem habentis • Cum 

'demonftratum fit neque maioremeiTe > neque minorem . Quod 

crat&c* 



TroftftiO XIX. 

CViufcunque minoris portionis Sf^an^ar fiiperficie^ fqualls 
eftcurua?fuperficieicyiindricm:^iiue^^ de« 

faiptiu 




Uher Triwms^^ fi 

fcripd i & eandem akkudinem ctim ipfa poitione Jiabentis • 

£fto minor fphdsre por 
tbABQ&portiocylin- 
dri FD£G;circa intcgram 
(ph^amdefcripti> cande 
tamen altitudinem HB cu 
ipfaportionc fph^rica ha* 
bcntis. Dico fj^^icam 
fuperficiem ABC aequa* 
lem eife Itipcfficiei cylin<- 
^lriFDEG. 

5i enim non eft eqnalis^ vel maior erit vel minor * 

Ponaturprimummaior; & ipfi fphfricf faperficiei ABC. 
conftruatur gqualis (yt in pr^edenti) cylindricu FLMG: fe<$o 
deindearcu AB bifariam > & portioncs eius iterum bifariam,& 
(iciemper^circumfcribaturarcui ABC figuca. multorum late-^ 
rum I N O P Q»jcerminata ad diametros , quj ducunturper pun 
d^ A & C « Sitque per pr^i^fbm bifedioQem arcuum , femi- 
la tus R O minus quam rec^ta D L , vt propterea planum S T, du- 
<5himper O^ cadat intra pun(5ta D^ & L« Qucnudmodum in 
pr^cedenti&cCoQuertaturdetndefiguravniucrfa circ^ OH, 
& cx conuerfione figur^ I N OP Qtjaicecur poitio folidi fph^- 
ralis fub conicis fuperfjciebus contenta. 

lam fic « Quia fph^rica fupetiicies A B C . ^ualis eft per co 
ftrudhonemcylindricae FL M G , nuior eadem erit quam cy*' 
lindrica FSTG, & muit6 maior quam omnes conicae INOPQi 
multoq*>etiammaiorquamomQesconicae AVNOPXCQuod 
eft abiurdum, &contra principia Archimedis • 

AfMmffimMs^yUndricdmfMperficitm FSTG mdiorem effe 
$mnibMs c$mcif INOP SI^Hh emm fdfei . N4m exr3.M4*& 
j s .hMtMsceBgipotefi ^eemcdU I N O P ^jtqudes eJfefMfetfi^ 
cicicjUndeicMccnt^mtAinterfldHMmST:i &fUnsimqM$d dn* 

gaetMTferfMn^MlSss. 

AffMjMffimMsetidm^dn&MtdngemfAV.^enicdmfMfirfi^ 

ciem^ 



t§.quim 



ti 



i,2 l)e SfhiAfAi tf fMils ffhAralil. 

ciem ^qufp i Imed IV y maiorem e/e quam ilU qmafit Uned AV. 
^odquidem demonjhdttir apud Archimedem dd Profofitionem 
jj. de Sfhfra & cylindro . Sed(^ ex noftris deducifotefi . Nam 
rectanguhimfrofriumfuferficiei , quafis i lined IV , m4UMs tfi 
duam reiidngulumfrofrium ilUusqudfit i Uncd AV . Conti^ 
netur enimfub Uneis maioribus . 



N^E 




Ponatur 
deindo 
fphserica • 
fiipcrfici 
es portio 
nis ABC 
min.qua 
cylindri 
ca FDE 

G. 

Fiatvt 
cylindrica F D £ G ad fphaericam fuperficiem ABQqua? minor 
pionitur, ita FH ad HM . &centro T iemidiametro autem HM 
fiat hemifpha^rium OQP,, circa quod intelligatur cylindrus OL 
NP Intraarcumautem ABC figura infcribatur multorum late- 
rum A V B X C per continuam bifedionem arcuum ita vt late 
ra ipfius non tangant feniicirculum OQP, & conuertatur vniuer 
fa figura circa axem BT • Intelligatur autem radio TZ(qu¥ t e- 
da perpendicularis fit ad vnum latus figura? infcripta? ) defcribi 
fphaeram , quae tangat fingula figurae A V B X C latera , & cir- 
cahuiufmcwii fpha^ram defcriptus concipiatut fuus cyiindnis 

\ lam fic . cy lindrica fupcrficies FDEGper conflrudlionem efl 
ad fphsericam ABQ vt FH ad HM , hoc efl vt FG ad MI . hoc 
ex 6 hw efl vt re(5hinguIumFE ad re(5kangulum MN , hoc efl: vt eadem 
^^ • cylindrica FE, ad cylindricam MN . Erit ideo fpha^rica fuper- 
txfUcd- ficies ABC «qualis cylindrica? M N ncnope minor cylindricaj 
turinfia j ^ ^ hpc gft minor omoib. conicis A VBXC jquod cfl abfurdii . 

AJfum^ 



tJbulni 



>^ 



UherTmmti 

Affkmffimms cyliBdricdmfiiperJiciem *• ^qudim effe emBit. 
€9n$cisAVBXC. Sljipdfdset ex demenBratis . Smne enim tam } U toi 
cylindrus ^s^uhn ornnes illd conicd eiufdem dltitudims HBi f^ ^- 
circk edndemffhardm > ^ defcribtmtur . 

Conft^ t ergo fuperficiem ABC ^qualem elTe cylindricas DP 
GE*cum demonftratum fit nequemaiorem efTeinequemino-- 
rem. Quod&c- 



CoroBdrium L 
Exfrimd dudrum prsmijfdrum Profofitiemum 
fdtetfuferficiem integ^dm ffhdrf , dqudUm cffe 
fuperficUi cylindri fibi circumfcrifti , (jr eiufdem 
cum ipfaffhdtra alti tudinis . 

Cum enim hdmiffhdriiim ABC fuferficiem hd^ 
hear dqudlemfuferficiei cylindri AEHCj cjr item 
hemtffhdtium dlterum AD C ^fuferficiem hdbedt 
dqudlem fuferficiei cylindri AFGC^ erit coniun^im totdffhdtm 
fuperficies fqudUsfuferficieicylindriJFEHG i exceftisfemfet 
hdfibus . 

Corolldrium U. 

MdnifeStum etidm efi ex vltimdfrefofitioney 

fufcrficiem mdioris ffhdrd fortionis , dqudlem 

efifefuf erficiei cylindri edndem cumfortione dL 

titttdinem hdbentis y (jr circi edndem ffhdrdm 

defcriffibilis . 

Cumenimintefftdffhptf fuferficies dqudlis 
fitfuferficiei cylindri JDGL , dr demonfirdtum 
fitfuferficiewftgmenti minoris ABC pqudlem effofuferficiei cy 
lindri EDGF , eritreliqudfuperficiesffhard AHC , dqudUs r^- 
UqudfufcrficieiEILF . ^odofortebdteSrc. 





ffdiei. 



Trofopsio XX. 



«. », ^ 



O VfKilides fphaer^quacirqtU cft maxiou cuculi in 'eadem 



fphxra defcripcibiiis . 




14 De SfhM, ^f<4uUsfffhdraliS. 

Sit ipbsera ABCD cuius diainettr AC; & cir* 
ca ipfamintelligatur cy liiidnis eiuTdein aldtudi- B ^ _i ?_? 
nis££HG. 

Dico fuperficicm fphfra? quadruplam efle 

ximi circuii in ea delcripdbilis . . _ _ 

Superficies enim cyliadri F£HG fine bafi« ^ I> 

bus , eft ad circulum (u? bafis circa FG, fiue cir- 

4^hu$Ms. ca A C defcripmm,vt EF a<J quar. partem ipfius FG , hoc eft vt 

FG ad quar. panem ipfius FG ; boc eft quadrupla • Propterea 

exfrim. ettam f uperficic» fpiMie^ae,qiiaccyiindricae eftaequalis «qiia* 

pZtd^ dfuplaeritcirculicirca ACdeferipti, quiin fphaera oiaximua 

' eft. Quod&c. 

AUter. 

SfhfricAfufitjpcits ABCD fqudUs tB cyUndricf FEHGicj^ 

Undricd vero FEHG 4dcirctdMm^^M$MsfemididmtterJ!$ AC^efi 

1,bu$mr ^freSfdngulumfer Axem.£0 y dd ijtudrdimm ex ftmididmetr^ 

lACy ttemft ddquddrdtnm EG i (^ide^ffMdUs : fr&ftertd etidm 

ffhfricdftifirfcies fqudlis.tritxtrctth cuiusfemididmeterfit 

ACiergoquddrufld erit circuU cuiusdidmeterfit AQ.Ssfd&c. 

JPropofith XX 1. 

CVtuicuttq; portioms^f{^erg fuperficies aequalis eft circu^ 
lo,cuius femidiameteracquaiis fit liaeae quae ex polo 
poitionisperducitur ad circuium, quiin eiufdem portionis bafi 

Jcft. 
£ft o fphaerac portio fiiie ininor ^ -. 

filietnaior ABC.cuiusexpoio ' j 

du6lafitrea»ABt Dico fi^cr- ^j 

ficiem portionis aequalem efle 

circuloquifitexABtamquam fe 

midiametro • 

Cum enimx]uadratum ABae- 

qualefitre^nguloDBEc^dKpliiai,ae<]piale eritdc reAan« 

guloGFIHyquodidemeftacre^ngulaiADBE* Prppcetea 

circu* 





UierVrimm i jg 

ctrcuIusexABaequalisericTi^lkieicylihdri» cui per axem 
Ct redang. GFIH , Sc ideo ae«|uali$ edaaKruperficiei ipliaed* 
caeporttonis>ABC. Quod&c. 

Tridhac TheoremdtA, qH/tfiqiumm-t exArchimede de/km- 
ftMfHntiqindq$udemfecimMsneU&ar\£tthu»edm ^uiire ^«< 
^eretHrtfedvmuerfmk'4ncd$ariM4mi»JiHliheU» idierei, 

Profofitio XXIl. 

SlntduoconireaiABCDEF. Sitq;cun»Bconi ABC f». 
periicieifquaUscirculbaDFjneinpcbafisalterius coniD 
E Fj redte verd IH , quf excentro I du- 
citurpcipendiculariter adJatU5AB,aN 4(1 

quaIisfital£itudoEL:DicoconosABC» B 
'DEF,efle?quaIcs^ . h/\ ' 



NamaltiudoBiadaltitudinein E 




eftvtBIadIH(ob^ualitatem)fiaevt^ I CD 
BA , ad A I , nempe vt curua fuperficies 
ABC ad batim AC ; fiue vt bafis DF ad bafim AC. reciproci. 
Qjare sequales erunt coni ABC , DEF . Qupd erat &c. 

CorolLiriHm . 
«iHCfdtet qtihdfcenHS AUqms , pntJt DOF . hdfim qmdem 
b^beHt DF tqHalem CHrH^fuferjjuiei ABC , HltitHdiMem vero O 
L non fqHoUmperpemlifuldri IHi Itafore couHm-ASC ndcoMS 
DOFyVt eft IH dd OL . N4m conns DEF Jtd conum DOF. eftvi 
EL . ddLO . £rgo (fnmftis dntecedentiHm dqH4iihHs) comu A 
£C 4dconHm DOF , erit vtlff idOL . 

Trofofim XXllL 

SI fucritrombus folidus A B C D , ex dubbus conis rcais 
compofinis ; Sitq; conus EFChabens bafim EG aqualein 
ft^rfidei curua? altcrius conorumrombi , puta, B AD; aldmdi- 
nemverdFHaequalemredaeCL,(pi^quidem exvertice reli- 
qiutcomfiCDducfturperpendicukaiterialatHS AB produ^ 

H s alte- 




j^ De Sfhdra, (g folidis fphAralik. 

alteriusconiBAD. Dicorombu 

folidum ABCD^qiialem efle co- 

noEFG* 

Ducatur I N peipendicularis ^ A 

adAB. Iam>conusBCDtadco- ^* 

xiumBAD,eftvtCIadIA;&c6^ * 

ponendo , rombus ABCD ad co- 

numBADeftvt CAad Alsfiue 

. vt CL , ad IN . Conus vero BAD 

jp#r C9f. adconumEFGeftvtlNadFHiergoexaequorombusABCD 

ffdui. ^ conum EFG eft v t CL ad FH . Ergo ^qualis . Quod era^&c 

Trapifitio XX />. 

SI fuerit conus fiue rombus 
liblidus ABCD fedusplano 
£Fa4i>afimparallelo. Intelli- 
gaturque cx integro folido ABC 
£) ablatus rombus foiidus EBF 
D . Dico reliquum folidum ex 
, cauatu AEDFC quodfupereft, 
eqiiak effe cono cuidam M , cu- 
ius bafis M fit aequalis &ufto cur- 
ua?fuperficieiconicse AEFC in- 
ter plana ELT , A C , intercepta? , 

altimdo vero M fit an^ualis perpendiculari DI > quas a veruco 
ablatirombiDduciturialatusBA. . 

Intelligantur tres coni a?quealti L, M, N . quorum vnicuiquc 
altitudo fitaequalis xeA; DJ s bafis vero coni L fita^qualis curug 
fuperficieiconiE&F. at bafis M aequalis fit fegmento conicae 
fuperficiei ittter plana Ei^ , AC intercepto : coni tandem N ba- 
fis asqualis fit vtri^ue fimul pra?dii5tis bafibuss fiue ( quod idem, 
cft) integr? fupertoei curu<g coni ABC . 

ManifeftumeftquodintegrumfolidumABCDasquale ^t 
<:QaoN (peralterutraoi prxcedentium duartun Propo£ j fed 

ctiam 




Uhr Frimus . 17 

edamduoconiX &M fimul fumpti aequalcs funteidemcono ^j^ ^^^ 
N. ergo intcgram folidum ABCD aequale erit duobus conis L ^imi . 
& M Smui fumptis . Demptis itaque , rombo EB/^D , & cono 
L s qui per prascedentejn funt aequales » reliquum folidum exca- 
uacumAED/^CfquaieeritreliquoconoM. Qupd^rat &g 

"Profofitio XXV 

SI ex cylindro auferatur conus eandem ipfi bafim,.&ean- 
dem altimdinem iiabens , erit reliquum excauatum foli- 
dum,quodexcylindrofupereft, »)uale cono cuidam, cuius 
bafis aequalis fitjiuperficiei curua? cylindri , altitudo vero a^qua- 
lis femidiametro bafis ipfius cy lindri • 

Efto cylindrus , cmus re<ftan^Ium ^ « 

peraxemfitABCD.&exipfoauteratur A S p yiv 
ponus BEC, vt diiftum eft . Sqmatur aiji- j^Nj /^"^l^s. 
temaliusconusi^IL^cuiusljafis-FLae- BON C^ ^ ^ * 
qualisfitfupcrficiei cura; cylindii, alti- 
tudo «jualis red? NB.hoc cft femidiametro bafis cylindri . Dt 
co rel iquum ex cy lindro folidum^ dempto cono BEC , rauale 
cflecono/^IL* 

Secctur BN bifariam in O. Conus ergo F\L ad conum BE 
Q rationem habet compofitam ex ratione altitudinum H I ad 
BA, hoc eft NB ad BA,&exrationcbafium, hoc eftbafis 

quaecirca/*LadbafimquaecircaBQfiucquodidemeft/uper* 4, hmtm 
ficiei cy lindricc ad bafim propriam quae circa BQhoc eft,Iine? 
AB ad BO • Erit crgo conusVlL ad conum BEC, vt NB ad B 
O, nempc duplus: folidum etiamcylindricum excauatum,dem- 
pto cono BEC, duplum eft eiufdem coni BEC . Propterea fo- 
Jidum cylindricum cxcauatum.aequaic erit cono F\L , cuius ba- 
fis a?quatur fuperfici^i cylindri, aliitudo vero aequalis cft iieaii* 
diametrobafiscylindri» Quod&c 



Pw- 




3l !DeSfhiatii^fgfiMsffh»M: 

Fropofim XXVL 

SI ex cono conus auferanir eandem habens bafin aldoidU 
nem vero minorem, erit excauatumfolidumcooidi^qiiod 
relinquicur » orauale cono cuidam , cuius quidem bafis aK|uaJ[is 
fit curua? fuperhciei totius prioris coni^altitudo vero ^qualis per 
pendicuIari>quxexverticcabladconidemittitur inlatus maio- 
risconi. 

• 

EftoconusreiftusABCexquo aafe 

ratiirconus ADC,vtidi*aumdft» Po- 

natur autem conus E/^Chabcns bafi n 

EG , equalem curu? fuperficiei coni A 

B C \ altitudinem vec6 Wf ^qualem re- 

d9.Dr»«qu( peiipendicularitet i^vertice 

abiaa coni cadit in latus AB . Dico folidum conicura eitcaua-* 

tum ADcBi dempto cono ADc , ^quale eflfe cono EF G . 
•Nam cum triangula BLA^BID, redangula fint, habeantque 

angolum communem A B L » fimiliaerunt • Sed conus EFG ad 

conum ADCrationemhabetcompofitamexratione bafium, 

nempecirculi circaEG,fiuefuperficiei curu? coni ABC >ad 
••5?ij?' circuhimcirca AC, hoc eft re(5l? BA ad AL ; fiue BD adDI, & 
h-fixtv . ^^ f ^tjQj^^ alti wdimim , nempe HF ad DL, fiue DI ad DL . Co- 

nus ergo EFG , ad conum ADC erit vt linea BD ad LD . Sed 
cdnus A B C ad conum A D C eft vt BL ad LD , & diuidendOj 
etiam folidum excauatum ADCB ad conum ADC eft vt linea 
BDad DL« Propterea conftat folidum excauamm ADCB 
^uale eife cono EFG • Qupd &c. 

Lemmd. 

Sidb eddemmdgnitiUUne A B dtu mdgmtMdines i$$£qudlet 

duferdntnr AC , mdior , & AD mincrfiierttq; DC^ nemfe differt 

tid inter dhldtds , dtqudUs difftremiffiue excejfui^quo mdius rr« 

fiduum BDfuferdt qudnddm mdgnitudinem E ^ Dico iffdm E 



Br 



I 



UhnFtimm. S9 

i refidm^ CB fqudem efe . 
Pdfet hec . Cmm emm maius reJiduMm DBfitferet 
tgnitudinem E excejfu DC ; fi excejfus dhycijisur^ 
erie reUqud CB fqudbs mdgnitudini E.Profteredmd^ 
gmitudoEfqudisefiminmreJiduc. ^odc^c. 

VrofofimXXVlL 

SI exconicoiruftoconusauferatur, quipro ba- 
fi habeat maiorcm fruft i bafim, altitudinem ve- A 
r6 candem cum frufto ; Erit reliquum excauatum fo- 
lidum a^uale cono cuidam» qui b afim.babeat aK]ua- 
lem fuperficiei curu? fi-ufti , altitudinem vero ^qualem perpen- 
diculari qu^ ducitur ex vertice ablati coni in latus alterum coni- 
cifiufti. 

Efto conicum firuftum ABCD, 
cuius maior bafis fit circulus cir- 
ca bQ. £t ex ipfo auf eratur co- 
nus J£C> cuius bafis fit idem cir- 
ctUus circa BC ; altimdo vero /*£ 
eademcumfi-ufto. Dicoreliquu 
folidum excauatfi dempto cono 
f£C,equale efle cono cuidam» 
cuius bafis aequalis fit curuae fu- 
perficiei conici frufti ABCD . alti- 

tudo vero fit linea £H • quae nimirum ex £ vertice ablatico» 
fii cadit perpendiculariter in Ab latus conici fi-ufti • 

Infaibatur alius conus AfD habens ba&n circa AD» & ver 
ticem in F , Pucaturque A I paraliela ad J5 F , eritque tota I C 
«eqnaJis vtrique fimiu femidiamecipi>afium,'4iempeipfi£A» 
'tpCui; FB. Fiu deinde circa FB femiciicukis FOByin<iuo. 
appUcetorjSOaequalisipfii^l^fiae ipfi Ejt;€riu{i circulus ex 
lemidiamctro J^O difierentia inter duos circulos»qiioram femi- 
4iaiactiifiBC^^^Q^£ue/r^&jEA*iicm0e ^ifejrcnoa in- 




ter 



40 lye Sfh^df ^ foHMs JfhArdlih. 

ter bafes oppoficas conici frufti , hoc eft iriter bafes cononm 
i5£C, A/^Dj&pFopftrea conus cuius bafis fi circulus cx FO 
femidiamerro r altimdo vero FE , difFerentia erit , fiue exceifii^ 
quo maior cotius SEC fuperat minor em AFD • 

Ponatur reda quaedam L , cuius quadratum aeqwjJe fitre- 

dangulo ex AS in lC , eritque circulus,qui fit ex L femidiarae- 

frof.xg. tro,aequalis conicae fupcrficiei fiiifti ASCD . Dcmittatur Jdeni- 

Muj . que ex F reda -FM. perpendicularis ad A1& ,8ccxE rc(5la E N 

paralL ipfi HM , eritque fada figura £HMN • parailelogranv- 

mum re^nguhim . 

lam cum propter parallelas HM , J? N,fint aequales anguU 
BA D, N£D , demptis reftis lAD , FED , erunt reliqui BAI , 
NJ?/'aequaIes;&ide6trianguIaB-rfI,N£/', cum redos ha- 
beant angulos ad I & N aequiangula emnt • 
j . Mum cum autem redbngulum BIC fimul cum quadrato Fl aequa 
^' ' Ic fit quadrato F B, vel quadratis jF O , O B, demptis aequali** 
bus B0| Fl. erit reliquum redbngulum B I C quadrato/* O • 
aequale * 

C oncipiatur iam conqs AFD detrahi ex conico fiiifto A b 
♦.*«- C2>, eritq;reIiquumexcauatumfoIidumdemptopraedidoco- 
^* no,aequaIeconocuidamcuiusbafisfemidiameterfitL,aItitu- 
dovcro/^M. 

lam : quoniam ob fimilitudinem trianguIoru,eil NF ad FE , 
vtBIad BA, hoceft(fumptacommunialtimdine)vtre<5hui* 
ulum BIC ad redtangulum BA in IC , hoc efl,fumptis aequa-- 
ibusjvt quadratum FO ad quadratum ex L reciproce, aequales 
eirunt coni reciproci quorum aker altitudinem habcat FE,& fe- 
midiametrum bafis FO ; alter vcro altimdinem habcat F N ^ & 
femidiametrum bafis L • Sed conus ille qui altitudinem habeat 
Ffi^ & radium hafis F Q» efl exceiTus inter ablatas magnitudi^ 
nes , nempe inter conos BEC , AFD ; Conus vei 6 ille qui alti-' 
wdinem habet FiV, & radium bafis L , eft excefius quo maius' 
»4. *ii- refiduum totius magnimdinis (nempe conus cuius altituiio FN^ 
^ • & radius bafis L ) fuperat quandam aliam magnitudinem, ncm- 
peconum^QuiusaltitudoiVMy fiucBH^ i-adjuiaucem bafisLs 

cric 




LiiirPrmifi > 4} 

crititaque haficmagnimdo, per LcmxMptxmiffvm$ cquaus 
mioorirefidiaoiergocoQUs prxdi&m$ cuius altitudo EH$it 
bafis circulus ex L aequaiis fuperficiei conici fi-ufti » aKjualis crxt 
minorirefiduo» hoc eftreliquo conici fiiifti ABCO.denqpt» 
cono B £ C . Qiod erat &c. 

Aliter. 

Sedt^mmnr Uem eBendere mimts Iditnefd demeuShMienei 
Jsfe^ile erit ex dipeutute mdtewt^i^ veriMs ex temtitdte iit-. 
genij. 

Sit eememnfrufiMm ABCD 
cuimsmdier bsfis BC^f^ *xiffi 
dstferdtnr- cenus B£C, dltitndi- 
ttem hdbens edndem cum frnfie ^ 
^fre bdfit mdiorem ipfins/rnfii 
bdfim . CemfUdtnr cenns BGC , 
snins ddtnm erdtfrnfimn » duHd- 
qne EH dd dngulos reifesiffiBGt 
fendtur JL medidfrofertiendUs inter GB, BF , m/f ,• cirenlns ex 
iLfemididmetredefcriptusy dqudlisfuperficiei ceni BGC .fidt pt, c«« 
tircdtLfermeirculns IMLiinqne dftetur I M medid frefortie» t*Mmi 
ttdUs istter G%A , AE , eritqi ckrculns exfemididmeero IMfd^ns *^ ^; 
^udUsfuperficieiconiAGDiJteUqnnscirculns exfemididme'. tJMmi 
sro MLfdHns , dtqudbs eritfuferficiei conic*frufii ABCD. (fi 
enm db d^ndUbus ^iqudUd demds reU^udfunt ^qudUd . ) 

Dico reUqunmfoUdnmfrufii conici ABCDidbldto eono BEC» 

dqnde effe cono ciuddmycnius dttitudofit EH ; bdfis vero dqud- 

UsfuferficieieonickiffinsfruHiihoe eficircnius exfemididme^ 
troMLdefcriftus. 

Cum ,n.duocircuU ex rddijs I M •, LM fdSi xqudles fint cit* 
eulo ex IL defcrifto^ dltitudo vnicuique eddemdfumdtnr EH^ 
erunt duo conifimnl (quorum sdtitndo communis EH , bdfes ve» 
vocircnUexrddys^IMtLM)dfudUseono, eniusdUitndo eddem 
£H,bdfisverocircnUisexILiiffeveroconus£qudUs eMf^Udf %tiMm 
eonico *BECG^emfto cono BEC, ergo dno iUiconifqudUs erunt 
foUdoBECG. ProftereidbldfisvtrinquedqndUbusconis,nem 

F fi CO' 



\ 



4"^ ^eSphira, (^foU^sfphdraliL 

fe f0n9 , CuiHs bdfis ex IM eHy dUitmdo EHy (jr cono AGD (fnnt 
eMm dqudlcsfeY z 2 . huius) remdnebunt fqndUd yfolidnm nem^ 
ve excdudtimfrujli ASCD, detrdito cono BEC , & conus cuius 
dltitudo EHjldjfs circulus ex LM rddiofdSfuSyfui quidem fqud* 
lis effuferficiei conicffruBi ABCD . ^odo^. 

Definitio . 
Si ex cylindro cyltndrus duftrdtur dquedltusy ^ chrcd eunde 
dxem dcfcriftusyfolidum excdUdtum quod relinquitur , Tubum 
cjlindricnm dffelldbimus . 

Trofofitio JCXVHL 

CYlindrus ad tubumcylindricum 3equealmm,eft vtquadra- 
tum femidiametri bafis cylindri ad r e<5lang ulum bafis ip 
fiiistubicylindiici. 
J: Efto cylindrus AB cufvs axis 
CD . Tubus vero cylindricus EF 
(dempto nimirum cylindro GH) 
?quealtus fit cum cylindro A B . 
Dico cylindrum AB ad tubum E 
F efle vt quadramm AC femidia- 
mctri bafis cy lindri , ad redangulum EGI,nempe ad redhngu- 
lum bafis mbi , hoc eft quod fit a differentia EG . & ab aggrc- 
gato Grl femidiametrorum bafis ipfius tubi . 

Nam cylindrus integer EF ad cylindrum GH,eft rt quadra* 
(utn EL aa LG • quadramm . Et diuidendo > Tubus cylindricus 
EF ad cylindrum GH eft vt redangulum EGI ad quadratum G 
L. ScdcylindrusGHadABcylindrumeftvt quadramm GL 
ad quadratum BC • Ergo ex aquo crit mbus cylmdricus EF ad 
cylindrum ABvtreaanguIumtGJadquadratura AC* Con- 
Berte0doigiturpatetquodpropofitum crat. 

Fropofah XX IX. 

DAt9 figurae folidae romndae figuram irifaibercaltcramque 
circumfcribere excylindris sequealtis, ita vt defcripta- 
fum di£Fercnt ia minor fit quolibet dato folido • 

Efto 





i 



H 




£fto cylindnis AJHC 
D>cuius axis EF : datoq; 
Intra cyiindnim Jfolido 
AEDcirca eundem axe 
EF reuoluto» fiue hemii^ 
ph(riu»fiue conus» vel co 
noides (it » oportet ipii fo 
lido AED duas figuras 
excylindris aquealtisco 
pofitas f alteram quidem 

infaibere » alteram vero drcumfcribete ita vt circutnfcriput fu^ 
peretinfcriptam minoriexce^ qua Gt quodlibet datum foli- 
cinmK^ 

Secetur hifariam cylindrus AC plano HG ad axem EF ere- 
Ao s iterumq; cylindrus HD bi£iriam feCetur plano I L i & hoc 
fiatfemperdoneccylindrus aiiquis puta AL minor remaneaC 
quam folidum K • Tunc diuifo toto cylindro AC in cylindro^ 
sequealtos ac ipfe AL , driantur in folido AED fedliones M N» 
OP. QR • Concipiamus fuper vnoquoq; circutonim MN, OP , 
QR , duos cylindros ^ altenim quidem verfus E » alterum autem 
verfus partes F conuerfum « Eruntq; omnes (imui cylindri qui 
verticem habem verfus F » an^uales omnibus (imui cylindri^ 
verticem verfus E habentibus(cum (inguli fingulis ^quales (f nt) 
Ergo fi onmibus cylindris qui verticem habent verf us E, addas 
cylindrum AL»fuperabit iam iigura circa folidum AED defcri-* 
ptayfiguram eidem infcriptanvaifferentia ALs Nempe minori 
exceifu quam fit folidum K. C2}K)d er at 6k. 

CorolLiriim . 

HimefMtt i/Mid ddidfgmrdf^Udd^ fiue btmiffh eriumJitJsM 
€§Ms jfime c0Hdides <irc Jipfi dm( fgwffolidf ex cylindris aque^ 
diisc^mpefiu dlierdinfcnhifetefi^dlterdvero circumfcribiiitA 
ntt differeniidiuter ddtdmfoUddmfigurdm,dr dcfcriftdrudlteru 
trdm^min9rfitqu$lAet ddtofoUdoK y 

Differentideniminteffigurdm ddtdm (§* dlterdm de/criftd^ 
fumminorvfiqieritqudmdifferentidinterdefcriftds(efi enim 

F M fdrs 



-«>w^ 



fmimfdem) erg$mmlihmin0Tq$timfiUdmm K. 

Tropofitio XXX. 

SPbofra quadrupU eft coni cuiufdam» qui quidem conus bi* 
fim babeat ^utletn maximo fphas^ circulo » aldtudinea 
vero eiufdem iphaar^ femidiametro aequaiem • 



Efto circulus 
cuius centrum 
i\;.quadratum 
ipficircumfcri 
ptum (it BCD 
£ ; iuadifque 
5A, AD- 
conuertatur fi 
gura circa^ 
axem F G ita 




E 



j) --.I-- 



vt a quadrato 

fiatcylindrus, 

a iphasra circuhis ;dtrianguIo £ A D, conus E A D • 

Dico fphan-am quadruplam eflfe coni EAD • Niii enim qua- 
drupla fit , non erit ha^mifpbao-ium ^uale folido , quod de« 
icribitur a trianguio EHA . circa axem FG. conuerfo (cum hoc 
folidum duplum fit coni E AD • ) Erit itaq; hemiiphatrium vel 
maius , vel minus folido tr ianguli £ H A • 
Efto primum maius,fi poteft elTe ; fitque exceifus asqualif fo^ 
htlm^' lidok» Infcribatur in hemifphaerio figura ex cylindris a^qudal- 
tis conftans ita vt ab henufphxrio dehciat minori defedu qua 
fitfblidumK. Eteritfigurainfcriptaadhuc maior quam foli- 
dum trianguli £H A. Secetur etiam axis AG in tot partes aequa^ 
4es in quot fe(^us erit AF • Dudifq; per punda fedionum pla* 
nis ad axem ere€tis , inteliigatur in folido trianguli E H A « in* 
fcripta figura ex tubis cy lindricis aequealtis conftans , quonun 
vnus fit , cuius fe&io eft redangulum HO • 
lam cy lindrus IL ad tubum cylindricum HO>eft vt quadra- 

tum 



I •'*« 



ciimlPadredangulumMON.. Sedquadratum IPaBqnale cft 
rcaanguloFPG,ncmpeipfiMON(namF P aequalis cft re^ae 
BR , fiuc ME, fiuc MO , & rcliqua PG reliqu? ON ) crgo cylin- 
drus IL aequalis e A mbo cylindrico HO • Hoc modo proccdcn^ 
dooftcndunniromncscylindriinhaemifphcrio a^ualcs omni- 
bus mbis in folido niangyli £H A . Qu^re figura in hcmifph^ 
rio infcripta excylindris conftans,aqualis erit figur2ein folido 
trianguliEHAdcfcriptarex mbis cylindricis compofitf . Sed 
figura in hcmifphazrio defpipta maior ccat integro Iblido.tidan 
guliBHA. £rg6ncccirccftquodfigiurainfcripta.infoiido E 
HAcodcmfoiidomatorfit.parsfuototo, Qiiodclfcnonpo* 
teft^ 



ptfxt 
huims . 



y 


'-Vj 


t 


^ 


^ >f 




t/ 


^k 


\ /• 


J'^^ m 


Vtm 


1 3C \ 


/N. 






buiw 



Efto dcindc,ff fieri 
poteft ., hemifphan-iu 
niinus folido triangu 
li EHAi fitq;dcfe^ 
dtusasqualis folidok 
Circumfcribatur 
ipfi hemifpharrio R-^ 
gura folida cx cylin- 
dris 2equeahis con- 
fians, itavtcxceflus 
figura? fupcr hcmif- 

pha?rium minus fit folido K. Tunc enini ciccumicripta figura . 
adhuc minor erit folido a ianguli £H A^ Concipiamus dcindc 
folido trianguli £H Aaliquam iiguramcflrc circumfcriptam c6- 
ftantcm ex tubis cyhndricis a^quealtisac cylindri cx quibusco^ 
ftat figura haemiiphfridcircumfcripta •. 

lam primus cylindrus HV figurae circa hcmifph^rium defcri 
pta?,cqualis cft primo mbo cylindrico figura? circumfcriptse fo* 
iido triangttli £H A ; nam & iftc mbus , cy lindrus cft H F . 

Secundus cylindrus G I ad fecundum mbum LM , eft vt qua fft ^s. 
dratum GN ad rcdangulum LTF , ncmpc a?qualis (quadratum *""'' • 
cnim GN , asqualc cft rcdangulo ONP, fiuc LT F, nam re<aa O 

Nrc- 



4i De SfhiTM» &fdi£srphArM. 

N rcdx BC^fiue L £ , fuic LT^xq ua&dt , & reli<]ua NP rdi- 
qu»TF.) 

Ergo omnes funul cy lindri figune circa hetniiphfrium deJcri 
pta^iioc eft eadem figura , anjudlis ertt omnibus umul mlns cy- 
iindriciscircafolidumaianami EHA dercripciS} cumi fif^uU 
iingulisxqualesfint. Scd hgura circa hemiiphxrium defo-u»- 
aminoreratiblido triangull EHA. hfccefie igitur eft quod i&- 
Udum ttianguli EHA maius fit » qaam<^ura libi circum&xipca . 
parsfuototo. QuodeOenonpoteft. 

HenufpKxriumigitirnequcmaiuStnequemiaus «ritfoUdo 
irianguU EHA, fed ipfi »quaIe,lolidum ver6 triangtUi EHA du 
plum eft coni EAD.ergo hemirphxriu duplum erit coni EAD , 
Sphxravero eiufdemquadrupla erit, Qupderai propoiitum. 

Hin6f4tetfph*rAmfHhfefquiaktr4MtJfecylmdTi, cmms hM- 
fisdqtiMsJit mdximoJfLer£ circuU, 4hitMiU 'ver^ di4mW» 
Jfhttr4d^ti4lis. 

N4mffh. ofieditttr t(ft MdftnimMADvt ^tdd^uBM^conusve- 
W EA1> 4d cylindru £BCD eff vt vtiS^d^. erge exdqMoffh^- 
r44dcylindrumEBCi) erifvt^-fda. NemftJtAJefqm^er4, 



DE 



47 



DE SOLIDI S 

SPHAERALIBVS 

LIBER SECVNDVS. 



Profofho Prims , 

ONVS quitibctcurafphxramdcfcriptu5>3e- 
quattscftconocuidain, quiba(im habcat a> 
qualcm vniucrfae lupcrficiei tircumfcripti co- 
ni accepta etiam bafi > olcitudtncm vero xqva- 
lcmradiofphaera:; 



Efto circa fphxram , cuius ccntrum 
A, defcriptus conus BCD, (qui vidcii- 
cetfpharamtangat & lateribus , & ba- 
fi) Ponaturq; alhis conus EFGi quiba- 
fim habcat EG xqualem tum cmiae fo- 
perficici, tum etiambafi coni BCO, 
altitudinem vcr6 HF habeat «quakm 
radlofph^AL. 




Dico 



4^ D^ Sfbdr4, ^ filidis ffhAfdlik 

Dico conos BCD » EFG a^quaics circ « 

Solidtim cnim conicum cxcauanim quod fit cx rcuoludooe 

i^.f ./«r trianguli CB A citca axcm IC» aequalc cft cono cuidam,qui i>a- 

^^ • fim habcat xqualcm curuat Ai{)trncieiconica? BCD r atcimdine 

v.cro xqualcm pcrpendicuiari AL , nempc radio fpha^ra? : Taiis 

crg6conusvnacumconofiAD(cumhabcant eandcm altitu- 

dinem) asquales crunt cono £FG; Quandoquideiffi conus EFG 

bafim babet vtriqs fimulbafi a?qualcm,alticudincm vcro alteru^ 

traexqualem. Proptcrea&conusBCD,quiduobdspratdi^ 

conisaequatur,aEquaiiscritconoBFG« Qgpd&c* 

utliter. 
Ducatur IM ^tquidifldns ipfi AL . &qu$^ 
u$i! uiMmanguhsCBldiuiditurhftrij^iUtftd 

BA , erit vt CB ad BI , itd CA ddAI. 
%. frim^ Suferficits^go ceni BCDfine idfij ddck 
f4ftii . culumfud bafis efi vt CB ddBI » nempe vt 

CA ddAI yfir comfenende , drfir cenuerfie^ 

nemrdtionis y erit vniuerfdfuferficies cani 

BCD cumbdfiy dd/ifperficiem eiufdem coni 

fine bdfiy vt IC dd CAy hec efi vt IM ddAL. 

PropteredfirecifTQceddhibednturbdfes , drdtitmdines , erii 

dgnms cuiusdltitudo AL , bdfisver}^ dqudUs vniuerfffuperficiei 

coni ECD cum bdfiy dqudUs cono cuius dltitudofit IMybdsisve^ 

Tojcurudtduiumfuferficiesconicd BCDyhoceBcono BCD(dqud 

les tnimfunt , conus cuius dltitudo IM , bdfis vero conicdfufer* 

ficies BCDy &4onus BCD .fcr 2 2 . huius .) 

Troftfim //• 

COnus quilibct circa fphft am dcfcriptus , eft ad fphafiram • 
vt coni ipiius vniucrfa fupcrficies accepta ctiam bafi » td 
fiiperficiem iph^(« 




£fto circa fphaeram ABCdefcripeus conus D£Fi 

iufmo* 



LiterSeeAnba 

luffflodi conum eflfe ad fphaeramtVt co- 
ni fuperficies vna cum bafir ad fuper fi« 
ciemfph^;. 

Ponaturconus HIL vtin praKed&* 
ti t cuius bafis fqualis fit integro peri* 
uietro coni D £ F vna cum bafi» akitu* 
dovero P I raualis radio fph^r^ O C > ' 
critq; conus HIL ^ualis cono DEF • 



m 







Pli 



i!^aturpercentrumOplanuinMN D < FB 
ad axem eredum> & in hemifph^io M 
CN concipiatur conus MGN • 

lam conus DEF ad conum HIL ( ob ?qualitatem ) eft vtto^ 
cus perimeter coni DEFD ad bafim HL > conus autem H I L ad 
conumMCN» (cumeandem habeantaltimdine) eft vt bafis %o.ef $• 
HL ad bafim MN> conus denique MCN ad fph?ra . eft vt bafis P* t^i 
MN ad fuperficiem fph^rf ( nempe in rationc fub quad nipla> 
c^are ex ^uo erit tonus DEF ad fph^ram» vt vniuenus perime 
terconiDEFadfuperficiemfph^f. QyiodSa:» 

• 

Trofoptio IIL 

C"^ Onus quilibet citca fphfram de/criptu$ > eft ad fphfranHvt 
J re^fhnguiumcQntentuinfublatere&femibaficoni tam- 

quam vna linea , & fub femibafi » ad quadratu diametri iph^'^ • 

Efto circa fph^tam > cuius diameter 
D£, deicriptus conus quilibet A B C. 
Pico conum ad fpheram efle vt re<5fcin 

J^uluidfub B AD tamquam vna iinea»& 
ub AD compraehenlum » ad quadi^a- 
D£. 

Curua enim fuperficies coni A B C 
adcirculum fuae bafis eft vtBAad A 
D, & componendo ^t totus coni peri- 

meter ad eundemcirculum bafis vtfiAi AD fimul ad AD s hoc 

• • • 

G eftvt 




i0 De Sfh^d^ ^ filidis ffhsralik 

eft vt redangulum fub linea B AD> & fuo AD ad quadratum A 
D ; cir culus ver6 bafis coni » ad circulam circa D B > eft vtqua- 
dratum AD ad quadratumDF » circulus deniquecirca D£ ad 
fphaerae fuperficiem , eft vt quadratum DF ad quadrattun DB» 
ergo cx aequo vniuerfus coni ABCAperimeter ad fuperficiem 
fphaerae ^hoc eft conus ipfe ad fphaeram pcr praecedentem^ 
erit vt reaangulum fub reda linea B AD , & fub AD » ad qua- 
dratumDE. Qupd&c. 

' CmUarium^ 

ProC^ToiUriofdteftoftenditofMm ^qMildierMm ddinfcriptS 
ffharam , cjfe vt g . dd^. . Poftto tnim latere A C. ff^ erit reStsn^ 
gulHmfnh Utere cwnfemibdfi y ^femihdft^j. quddrMum vrro 
SD zy.&qHAdratumBE 12. crgoconns Adffhdtdm eritvtjj 
dd12.ftHevtp.Ad 4^ 

/ Scholium. 

Pojfhnt hi^heoremdtdhonfAHCdfrofoni circdfolidorttm cit 
CHmfcriftiontm , (jr infcriptionem : qudlid/Hnt . 

SicircAj^hframfri/mAConcifidtHrj qHodftngHliifHisfdrAU 
lelogrAmmisffhdTAm CQntingAtyfttque einfdem AltitudiniSyErit 
frifmAAdffhArAmyVtferimeterbAftsfrifmAtis AdduAs tertiAs 
ftrifhArifmAximicircHliJfhdrA. 

Sivtr)>noneiHfdemsit Altitudinis; rditio frifmdtis Ad sfhfS^ 
rdm ctmfonetHT exfrAdiStdy dr exrdtidne dtitHdinum ; Altitn^ 
do AHtemfphfrA dJAmeter eft . 

Si cyltndro circHmfcribAtHrfrifmA , quod singHlisfnis fAr/d^ 

lelogrAmmisfHferftciemcjflindricontingAt;sintq; eiufdem aL 

titkdinis. Erit fri/mA Ad cyhndmm y vtbAsis Adbdsim: Hem^. 

fe y vtftrimeter bAsis frifmAtis , AdferifhAriAm bAsiscylbidri^ 

idem verum eftde oono , ^fyrAmidibus circumfcriptis . ^ 

Si veroprifmA , c^ cylindrHs non eiufdem AltitHdinisfHerint;' 
rAtio componetur ex rAtioneperimetriddperiphfriAmy dr Altitn^' 
dinis Ad Altitudinem . 

SiintrA cylindrnm iffcrihktHT frifmA eiufdem AhitHdinis ^ 
hAbens bAsimfoligonAm , regHlarim^ (^fariUterAm ; Eritcylin 
jkrns AdfrifmA , vtferifhipriA bAsis cylindri Adperimetrumfoli- 

goni 



g0niregMl4rism eidem circHlo defcriftiy qu§dh^dtlMitr4mul 
tiiMdinefMbdHfldfoligonibdsisfrifmdtis. ^dHerafHHi etidm 
elt c$no > (^ fyrdmidibHs infcriftis . 

^dndo Hero bdsis frifmdtis imfdtildierdfiierit , sine retHld^ 
^i^^ siHeirregiddris : Erit\cylindrHS dd infcriftHm frifmdy m 
ferifhfrid bdsis cylindri ddomnes sinns drcuHm kldteribns bdr 
sisfrifmdtisfHbtenforHm. DHmmodo nnllHs drcHsfcmicircHU 
mdior sit • Slndndo nero dfcHs d/iqnis/emicircH/o mdior sit ; (^ 
qHdndo jigHrdrHm dttitndo non sit eddem ^c^dlid hHiHfmodi^m 
tfi^demonlirdrifoJfHntfdcili quidem negotio \fedinftitHtmn m 
ftrnm eft non omnemfolidorHm infcriftionem^ (^ eircHmfcriptio^ 
nemfrofeqHiifedilldm, tdntHmyfHfcircdffhfrdmen^ Helin. 
trd iffdm i Proftered ddinceftnm reHertdmnr • 

"Pr^ofitio IV. %. 

SI circacirodumdefCTibaturpoIi^^ 
mcroparia,fiu$dquatcmario,fiue abinario menfurata, 
&rcuoiuamr figuracirca diagonaicln, eritlaaum fphacralefo- 
lidum aequale coixo cuidam qui bafim habeat aequaiem fupcr-* 
ficiei folidi » altimdinem vero femidiametro iphaerae aequaif • 



Hoc dutem qudndo numerusld 
terum menfuratHr iqudterndrio 
demonftrdtHmfuit db Archimede ^ 
Prof. ^fi.ftue mauis 2f* Ub.f.de Ci 
ffh.dr cylin. ^dndover^bldtern T 
numrHs ctidm g bindrio tdntum 2) 
mcnfurdtHryOftendemusftc , erit^ 
que demonftrdtio (exceftis qna 
de vltimo folido cylindrico dicen 
tur) eddem cnm ed qudm dffertAr ^ 
chimedes^ 

EftopoHgonum ABCDEFG habcL. ™^ . ^iu*.iu«e. 
tummcnfurata,vtinprimafiguni, Erg;d fcmipdigonum AB 

G a CDEF 





• • 



•'\ 






5^ UJeSphdra, tffoUdisjpb^r^liK 

CDEFhabebitkceranumeroimparia» latuique vnum tanget^ 
circulum in pun&o T » atq; ideo cylindricam fuper ficiem in co« 
uerfionedefcribet. Intelligatur conu$ MNO» cuius bafisfit 
circulus MO aequalis vniuerfx fuperficiei folidi iphseralis ,alti«* 
tudo verd PN » asqiialis fit radio iphsene . . Dico fphan-ale foli- 
dlim aequale effe cono MNO . 

^ Rombus enim folidus fadlus in conuerfione figura? d triangti 

^hffAf lo ABQjcqualis eft cono cuidam cuius bafis ^quahs fit coni- 

'^' * ex fuperficiei defcripta^ 5 linea AB, aldtudo vero fit radius QR. 

Soiidumautemexcauatum fadum incohuerfione a triangulb 

BGQijapquatur cono cuidam cuius bafis a^qualis fit conica? fu- 

perficiei defcriptaf a linea BC altitudo vero a?qualis radio fphg 

tx QS . & fic femper procedatur • Vltimum denique folidum 

%^'ffsf cy lindricum excauatum defcriptum a triano;ulo C T Q^ a?qua* 

^^ le eft cono cuidam » cuins bafis asqualis fit luperficiei cylindn* 

cae d linea CT fady, altitudo vero asquali^ fit femidiametro cy^ 

liodri , QT ; Et fic de f olidis circa alterum hemifphaerium TF V 

defcr^tis . Ergo vniuerfum fpha^rale f6lidum,acquaje erit om- 

nibus pra^didis conis fimul fumptis ; ijfdem autem omnibus 

prardidis conis asqualis efl conus MNO (cum bafim habeat om 

nibus fimul illorum bafibus a»]ualem > nempe fuperficiei folidi 

fphfralis» altimdinemvero vnicuiqueillorum ^ualem, nem- 

pe radio iph^r^ . ) Propterea pr^di^tum foUdum fphxral&fqua 

lccritconoMNO. Qupd&c. 

. Propofitio V. 

SI circacirculutndefcribaturpoligonumhabenslatera nu^ 
mero paria , & conuertamr figura circa diagonalem : ha- ^ 
• bebitfadmnfpha^alefolidumadipha^amfuam eam rationl» 
quamvniuerfa foiidi fphacralis fuperficies habet ad fuperfi- 
ciem fphaerae • * 

Manente praecedentis Propofidonis confhiidione ; Eflo 
fphaerale fblidum cuius diagonalis , atque axis fil AB, ccntrum 
autcmfphaerac fit C • Dico f|^cralc folidum ad infcriptam 

fibi 



H n 



liief Switifim t 



fS 




ifphaeramef- 
fcyVt fuperficies 
folidi ad fuperft- 
cietn fphaerae.. 

Infcribaturju 
in hemifphaerio 
conusDEF,&po 
namrconusGIH 
cuius bafis G H 
aequalis iit vni- 
uerfaefuperficiei 
folidi f^eralis 
vtin praecedenti 
altitudo ver6 L I • . 

aequalis radio fpbaerae9& erit per praecedentem fphaerale £>- 
lidum aequale cono G I H. 

PropteraequaIitatemerg6,eritfphaerale foKdum adconu 
GIH vt fuperficies vniueiia fphaeralis folidi ad bafim coni G I 
H> conus autem GIH ad conum D£F (ob aequalem.altimdine> 
efl vt bafis circa GH ad bafim circa DF s conus denique D EF 
ad fphacram , eft vt bafis circa DF ad fuperficiem fphaerae(n6- 
pe in rationc fubquadmpla « ) Propterea erit ex a equo fphae- 
ralefolidumadinfcriptam fibifphaeram vtvniuerfa fphaera- 
lisiolidifuperficiesadiupetficiemfphaerac.. Qupd&c» 

Tropoptio Fl 

Q^ I circacirculumdefcribaturpohgonumhabenslatera nu- 
jpi meroparia» &conucrfiaU]r figoracuK^adiagonalem^etit 
fa6ium iphaerale folidum ad in£biiptam. fibi fphaeram vtam 
ilblidiad axem j^haerae » 

Manente praecedenttum conftrui5lione j efto fphaeraTe foli- 
dum> cuius diagonaIis,.acque axis fit AB^centrum ver6 fphaera? 
fitQ&diameterHI. 

Dico 



fer I $• 

p:fartis. 

iZ.f.fsr 
tis. 

frims f» 
fartis • . 



J4 De Sfhstr^ftfMkJfUrM. 

Dicofph^ 
ralefolidCi 
ad infcrip- 
ta fit)i fph^ 

rameflTcvt 
ABadHL 
Circum 
fcrifauur lu 
circa iphae* 
ram cyfin- 

drusNLM 

O.agancur . 

quc pcr extremitatesaxis A, B>plana ad axem cre<fta DG i EF. 

per extremitates vero dianietri HI. plana LM,hlb . 

Eri^perpraocedentemyfphan^^ile iblidum ad fphan*am vt fu- 
perficies^pha^ralis folidi ad fuperficiem Iphan^a? ; tioccft» ( fum* 
ptisa9qualibus)vtfuperficies cylindriDEFG, ad fuperficieni 
cylindri LNOM , hoc eft vt AB ad HI . Quarc fph«rale foli- 
dum ad fphaeram eft vtasds iblidi ad diametrum fphaerar .<^d 




Propofitio VIL . 

SI intra circulum defcribatur poligonum habens latera nil- 
meroparia,&conuertaturfigura circa diagonalcm', erit 
fphasra ad infcriptum fibi fphaerale folidum , vt quadratum dia- 
metri fpha^ra? , ad quadramm cateti poligoni • 

* 

Sitn xirc. cuius cent A, & diamet. BC poligonum reguiare» 
cutus diagonaiis fit linea BQ& conuertatur figura circa BCDt 
co fphaeram circumfcriptam ad inclufum fphaerale foltdum» efie 
vt quadramm AQad quadratum catcti poligoni AD . Ducatur 
enim D£ ex D . perpendicuiaris ad BC , & EF perpendicularis 
4 •fticsu ^^ AD,eruntq; in continua proportione quatuor red? AC, AD, 
A£,AF. ConcipiaturetiamradioADaliamf^aa-amddb}* 

bi, 



Sf 




UkrSiemubts 

bi, qiue fingulas conicas fupcrficies fo-^ 
lidi fphxralis continget ; necnon cylin^ 
driczxn^ fi quamhuiuunodi fphasrale fo 
iidumhabebit« 

Erit itaque fphm maior. ad fphasra 
minorem, vt CAad AFj minorverd 
i^haera ad fph^ale folidum , quod fibi 
circumfcribitur ( perpnecedentem J eft 
vtDAadAQhoCieft', vtAFad AEi 
Proptere^ ex asquo erit circumfcripta^ 
A^han-a maior, ad infcr^)aim folidum 
wharrole, vt CA ad A£mempe vt qua- 
oratum CA ad quadramm AD.Qpod 
crat&c. 

. IProfofiM VIII. i u 

SI circa fphaerale folulufti, nanm ck reuolutione pohgoni 
circadi^onalemreuoluti,fi)haa:acircumfcribatur, &ai-' 
t€rainfcribatur«Hab«bitcircumkriiHafphana ad folidum,dur 
plicatam rationem illius,quam habet folidum ad infcriptara 
4>h;)eram» , , 

Repetita figura Propofitionis praxedenris ; cum fit circum«« 
fcripta fjph^a ad folidiun vt quadramm CA adquadratuo^ ADi 
folidum ver6 ad infcriptam nbi minorem fpha^am,vt C A ad A ^* buwti 
D spatetrationemcircumfcripta? ff^aerse id folidum iphasrale 
dupUcatameireilliusquamfohdumhabetadinfcriptam fpha^ 
ram. Qupd&c. ' . 

"Pntifnk /X 



SI circa fplmale folidum , naium ex reuolutione pohgoni 
circa diagonalen^ reuoluti > fpher^^iix^ 
rainicribatur: Erit fuperficies folidi IpFcilMU mediaproponio* 
nalis inter fuperficies duarum fphaerat um « 

Manan- 



Ofiendi'^ 
tur in 6' 
tuiui. 




si De Sphxird, (f fiMs fpbdfaUk 

Manente figura » & conftrudfone 
praecedentium propofitioniini . Dicd 
tres fuperficies » nempe maioris fpbae-- 
rae,(blidi fphaeraiis,minorifq; infcrip- 
tae fphaerae » effe inter fe in continua 
proportione* 

Superficies enim circumfcriptae i^hf 
rae efl ad fuper ficiem uiicriptae, vt qua 
dratum C A ad quadratuin ADsfuperfiv 
cies autem folidi ad fuperficiem eiufde 
infcriptae fphaerae, eft vt reda C A ad 
redlam AD:£rg6 tres fuperficies prae- 
di£tzc funt in continua proportione ; & 
quidemperimeterfphaeralis folidi medius proportionalis eft 
interfupcrfidesduariimiphacfanim. Qupd.&c 

Prtf$ptio X. 

SI circacircuIumdefcribaturpoligontBnhabensIatera iiih- 
mero paria , fiue a quatemario , fiuetantum a binario men 
furata;&cdnuenaturfiguracircacatetum; Eritfadhitn fphae- 
rale folidum aequale cono cuidam » cuius quidem bafis aequa- 
Ksfitvniuer- 
faefuperficiei 
folidi fphae- 
ralis; altitu- 
do vero ae- 
oualis radio 
j^h^ae. 

Eftocirca 
circulum fi- 
gurapoligo- 
na aequilate 
ra ABCDE 
H.habcnsla 




tera 



tetanumeropariat&conuenaturfigunckcacatenm Il^orie- 
turq; folidum contehtum fub conicis , circularibus » & vna cy« 
lindrica fuperfide , quando numerus laterum a quatemario me 
furatur ; quando vero i binario tantumitunc erit folidum fph«* 
rale fub conicis» & circularibustantum fuperficiebus compras- 
hcnfumvDicova-umq;fphaeralefolidum»]uaIee(re conocui* 
dam MNO» qui bafim habeatasqualem vniuerfas folidi fphaera* 
lis fuperficiei » altitudinem vero rN asqualem radio fphaera? . 

Hoc oftendetur fimiliter vt propofiaone 4. faAum eft Nam 
conus qui fit d triangulo I A Q^ conuerfione circa axem I L » 
arauatur cono quibafimhabeat^qualemdrculo qui fitex radio 
lA» aiticudinem vcr6aequalem radio fphasne Qltquiaidem 
prorfus eft • Soiidum autem excauatunitquod fit a triangulo 
ABQ^arquale probatur cono cuidam » cuius bafis acqualis m 
fitconicasmperficieifiiaaeilinea AB»aItttildo vero fitQ^.ni«^ 
dius fphaerar» Vltimumdeniquecylindricumfolidum excaua- 
tum,fa£himatrianguloBQS(quando poligoni latera d quar 
cemariomenfurantur> alids cyhndricum folidum nullumeft) V 
afcquaturconocuiusbafis^qualisfitcylindric^ fuperficieifii^ ^^ 
dlineaBS.altinidover6fit QS^;&fic de altero hemifph^o. 
Propterea vniuerfum fphasrale folidum fquale erit omnibus 
praedidis conis fimul fumptis ; & ideo aequale erit etiam cono 
MNO ^qui omnibus illis fimul fumptis aequiualet;(quandoqui» 
dembafimhabetomnibusfimuliilorum bafibus aequalem ex 
fuppofitione ;altimdinem ver6 vnicuique illorum aequalem» 
nemperadiumfphaerae^) Quod&c. 

Profofitio XI. 

SI circa circulum defcHbamr ix>ligonum habens latera nu- 
meroparia»&conuertaturngura drca catetum, habebit 
fa^mfphaeralefolidumaditifCTiptamfibi fphaeram eamra- 
tionemyquamvniueria folidi fphaeralis fuperficies habet ad 
ft^rfidem fphaerae • 



Manente prac^coctf propofitioDis cpfinidtione^fto fphae^ 

H ralc 



5 8 DeSfba^d» tffiMsffhATdlik 

rale f olidS 

cuius catc-» 

taSy &axi9 

iitABiCcn- 

trum autem 

jQpihaerae !k 

C. Dico 

fphaeralc fo 

lidum adia 

fcriptam fibr 

ftStacramef 

fcvtmiicr 

faipfius foli 

-diliyerfici- 

es ad fuperfickml^liacrae^ 

Concipiaeur ctiim in bcnniph;unk)i^€Oca^ DA^&inttUij^- 
tnraliusconusPGHsCiiiQsbafisFHacqDalis (k vniuerfae fu- 
pcrficici folidi fphacralis>aitituda vjt^ro IG ^ualis radio iphae-- 
rae > & eric perpraccedcnccm ^haerale folidfum acquale i;oao 
f^GH. 

Propteraequalitatemcrgo>erit4>haeralefolidiim ad conu 
FGH, vt fuporficicsvniuerfa fpfa«ralisfolidi , ad baiiiii coni 
f GH , conus autem FGH > jul cooum DAE /ob aequalcm akt- 
mdincm)cft vtjbafiscirca FH«adiia£mdrcaD£;denxqueco 





iphan-ale folidum ad inictiptam fibi fpha^am^vt vniuerfa fpha^* 
TahsfohdifupcrfidesAdiupcffidcai(phaer:E. Qupd&o 

Frapififia XI!. \ 

SI ch*ca circulum dcKi:k)atfu: j>dl%diium liabens lateia 
numcro paria , Sicoaiahwar&g^^ 
bitfadumfphaeralefolidum .ad infcriptam iSMJ^ha^/aoi^ai eam 
Tadonen)» quamiiabet compofitaredalinea ex dxamietro fpha?^ 

.ne> & ex tettia proporttotMlj< iS^jKyf iamimasaa^ ijftm^d. 

" femi- 



^MM 



tikirSictmiiiii f| 

leinHatus poUgcniiyitafemilatusadaJyu^ 




Manente prae 
dentium propo^ 
fidonumcoftra- 
^^oneyeftofph; 
ralefoliddcuius 
cacetus, 8c axis 
£it AB \ centrum 
autem fphcere fit 
C* Fiat angu- 
lus CDE rc^aus, eritq; BE tertia proportionalis ad femidiame- 
trum CB> & lemilatus poligoni fi D . Dico fphaaak folidum 
adinfCTiptafibi fpharaeflc vtEAad AB;nci^vtcompofita 
exdiametro/ph«a?AB>&tertiaproportionaIi B&ad diamo- 
trumfph«-2e AB . Concipiaturcirca fph*ram defcriptus cy- 
lindrus FLMI,& perpun^ A; B5E .producanturplana FI,LM, 
GH , ad axem eredtei » 

Erit ergo , pcr prxcedentem , fphaerale folidum ad infcripta 
fibifoha?ram, vt luperficies folidi ad fnpef ficiewi fph«rat; hoc u.pri. 
cft , fumpris ajqualibus, vt fuperficies fyiindri I^GHL ad fuper- /*'*'' • 
ficiemcylindriFLMIihoceftvtlineaAEadAfl. Qupd&c ^W^? 

"Profofttio XIII. 

SI circacirculumdefcribaturpoIigonumhfl&eiiBlatieranu- 
mero paria , & conuertatur figiu-a circacttetum \ erit bJSSa. 
I^hg^ale folidum ad fuam fphasram , vt doc^adrata, nempe vt 
^dratumdiagonalisy&quadratumcatedfkfiid, adduplum 
quadrati dufdem catett , 

Eftocircacirculum, cuiusceittFum A, defcriptam poligo- 
num habcns ktera numero paria,& conuertatur figura circa cat- 
cecum BC : ia^oqi angulo xt^o ADE> erit f^r praecedentem^ 

' H a foH- 





iblidumfphierale 
ad fuam fphjeram 
vtEBadBGDi- 
c6 infuper folid& 
jfohaerale ad fuam 
iphaeram efle , vt 
quadratum ex A 
D, fimul cum qua 
drato ex AC , ad 
duplum quadrati 

CX AC. A r^ 

Nam EA ad AC eft vtquadratum DA ad quadratum A C j 

«ccohiponendo, eruntEA,& AC fimul,hocefttotaEB,ad 

AC , vt duo quadrata D A , AC fimul ad quadratum AC ; fum- 

#»f tff ptifque confequentium dupUs, erit EB ad BC (hoc eft foUduin 

^ -Iphaale ad fpharam; vt duo quadrata D A > AC fimul » ad dur 

plunlquadratiexAC. Quod&c. 

Trapo/tfio XJV. 

Q I intri drculum defcribatur poligonum habcns la««a «o- 



^ mcro paria, & conncrtatur figura circacatetum; erit fpb 
ta ad infcriptum fibi folidum , vt integra diamcter fphcr?,ad fc- 
cundam fmiul , & quartam proportionalium , m ratione lem*- 
diamctri fphaera ead femicatetum poligoni . 

Sitin circutocuiusdiameter AB po> 
Jigonumhabenslatera numero paria, 
& conuertatur figura circa catetii CD : 
Ducanturque perpendiculares D F ad 
reaaHE,& FI ad HDj &erunt qua- 
tuor lineae EH, HD, HF, HI, in conti- 
nuarationefemidiametriHEad femi- 
catenim HD . Dico fphaera ad infai- 

.ptumfoUdumcire^vtdupla HE ad v- 

tramqi 




Uher Stcundm 4 8i 

JRamqjfimdDH^HI. Veivtinte£ra<Iiamcterf^^ CI. 

latelligaturalia fphaora intra tolidiun infaipta. Erit ergo 
cxterior^haeraadinteriorem^vtEHadHIyfiuevtduplafiH vuims 
adduplamHIjinterior vero ^hm adfolidumfphaerale eft» *»9d<ci- 
vtduoquadrataexHD,adduoquadrata HD)H£»hoceftvt'''' . 
duoquadrataexHI»adduoquadrataexHI>HF, hoceft ( vtftrfrft. 
infira oftendemus ) vt dupla HI ad HI > HD j Proptefea erit ex 
arquo (phasra extenor ad infcriptum iibi iphaerale foIidum,vt du 
pla HE , hoc eft integra diameter fph»a?> ad HI » & HD fimul ; 
quse quidem funt fecunda» & quarta in ratione femidiam. i^h^ 
rae aa femicatetum poligoni • Quod &c. 

SB^ddMtemdJpmftmneH oftendtmMs. Dico vtduoqua- 
drata ex HI ad duo quadrata fimul HI> HF • ita eiTe di^Iam H I 
adHI,HD, 

Nam ob angulum redum H F D , erit vt quadratiun F H ad 
quadratum H I , ita redtaDH ad HI ^ & componendo» fumpcLt 

3ue confequentium duplis, erit vt quadrata FH> HI, ad duo qua 
rata ex HI , ita duae redtae DH, HI, ad duplam H I • Conuer* 
tendoereo, eruntduoquadrata exHI, ad duoquadraiaHI> 
HF.vtdupiaHI^adHIsHDfimuI. QuoderatfiK. 



Vrpfofttio XV. 



s 



I circa circulum defcribatur poligonum habens latera nur 
O mero imparia, & conuertatur figura circa cacetum polieor 
ni, erit fa^tum fphasrale folidum asquale cono cuidam> cuius ba- 
iisaequalis fit vniuerfaei fuperficiei folidi ^ altitudo vero radio 
fphan-aefitxqualis. 

Efto circuli centnim A>polig. ver6 perimeter BCDEFGH* 
£tfintIateraeiusnumeroimparia;conuertaturque figuracirca 
catetum BI ^ vt oriatur folidum fphaerale contentum fub conicis 
fuperficiebus vnicoque circulo circa diametrum £F defcripto • 
Ponator iam conus 1 M N > qui bafim habeat aequalem vniucc- 

&fuperfideifph2nd[isicdidijaltiind^i^ OM agqualem 

radio 



radio fyhxxx AI . Dico folidum 
fplmalc dequalc cfTc cono LMN.P 

Agatur pcr ccntrum fphaer? pla 
num PQad axcm crc^dhim , quod 
tranfttcrsey fccabit aliquod latus^ 
poligonifpota CD * 

Erit iam rombus folidus , fadus 
a conucriione triang. BCA , aequa 

%ippaf ^^ ^^^^ ^ ^^^ baiim habeat arqua- 

11/, km conic« fuperfidci h&x i li- 

nea BC ; altitudinem autCm Mua- 

ktn radio lph?r? A R . SoUdum 

vero conicu excauatum quod fic 




ex gyro trianguli CP Ataqualc erit 

co no» qui bafim faabeat ^ualem fuperficiei, qu^ fit a linea C P^ 

• ••• \ f J*.# 1__ A O ^ I • I 



If/. 



rf/. 



altimdinem vero ;qualem radio iphgr^ AS • Spiidum quoquc 
^ excaiiatum t &dum ex rcuolutione trianguli PD A > rauatur co- 
no, qui bafim habeat gqualcm fuperficiei coniceqi^nt a motu 
lincf PD^ altimdinem autem ; qualem radio fph^r^ A S • Ea-- 
demprorfumeodemmododicunturdefolido conico cxcaua-- 
tOjfadloiorianguIoDAE; &de vltimo cono fado dreuolu- 
tione trianguli EI A ^ Pf opterea totum fpfajralc folidum ?qua- 
le erit omnibus pr? diiSlisr conis fimul fiimpris y vel cono L M N , 
qni omnitnis iilis pr^didis ^quiualct : (faabet enim bafim omm* 
busfimulillorum bafibus ^qualem, akitudiAcm vero ^qualcn^i 
ymcuiqiillorum.^ Quod&c# 

Sciotium . 
AttuUmusinhdcPropoJitioneTheor.23,24.(^2j.p. fdrtisi 
^dm exgyro tridnguli BCA oritttrromhtisfolidMs vt in 2 j.ffdr 
tif . Exgyro tridngnli CPA orittirfilidttm qttodidm exedttdtH^ 
qudle relinquitur Jiex cono dufcratur YOmhusfolidus : vt in 2 4. 
*f.fartis . Uenique ex contttrjione tridnguti DPA oritttrfolidum 
quodddm excdudtttm hdbens bdjim circularem P^ qualerelito- 
quiturjicxfruftocomc«cotouj duferMtxr hdbens bdftm edndem 

cum 



Uher SecuniHS . 6s 

^um mdimre bdjifrufti cmci ^ dltitudintm e^mqUi fJtndcm vtin 
rrpf^zj.f.fdrtis. 

Fropofith XFIL. 

SI circacirculutndefcribaturpoligonumhabens lateranu- 
mero imparia, &conuertatur %ura circa catetiim '5 habe- 
bit faiSum fphgrale jfolidum ad infcripram fibi fph?raixi,cam ra- 
tionem quam vQiuerfa Iph^ralis jrolidifuperficies habet > ad fu- 
perficicmfphjrg. 

Manente prfcedentis propofitio- 

snis conftrudione , fit fpherale foli- 

(dumcuiuscaterus , fiueaxis fit AB> 

ccnttxraverofphaerae fitC. Di- 

tco jphaenale folidum ad infcripta 

ilHlphaeram^fie^ vt ipfius fohdi 

iategra fuperficics ad luperficiem 

ijphaerae-»' 

Concipiatur in hemii^haerio 

;Conus DEF ; & intelligatur conus 

•GHlcuiiisbafis GI jqualisfit vni- 

ajcrlae fuperficiei folidi ^^haerolis^ altimdo vero L H aeqaalis 

iitradioyJiaeraeai&eritper {Haecedentem, fphaerale folidu 

aeqHalecono GHI • 
Propter.aeiqualitatem erg6, erit§)hacrale fblidum ad corfu 

G HI, v t fiiperncies vniueffa fphacralis fof idi ad bafim coni 6 

HI > conus autem GJHU adxpo^m D£F ( ob aequalem altitudi- 

nem)cft vtbafiscircaGIyidfaaffmcitdiDP. conus denique 

DJE£»ad iphaeram^efi: , vt bafis circaDF ad fuperficiem fphae- 
, rac(nea^e inxatione fobquadrupfa. ) Propterea erit ex ^t(hk , 

i^haeralefoliddm adinferiptam fibi fph<iera,vt vniuerfa fpiiae^ 
.lulisXoJidifuperficicsad iiiperficiem fphaerac. Qupd&c. - 




* , 



Pro- 



64 



De Spkar^, & folidis ffh/arM. 
Fropopio XVI L 




SI circacircuIum4crcribaturpoligonumhabens laterann*- 
mero imparia»& couertatur figura circa catetum poligon^ 
habebitfadumfphaeralefolidumadinfcriptam fibi 4>haeram 
eamrationem ^quam habetiineacompofita ex cateto poligom 
& tcrtia proportionalium ( fi fiac,vt diameter fphaerae ad lcmi- 
latus poligonit ita femilatus ad aliamt ) ad diametrum fphaerae* 

Manente praecedentium conftruftio 
ne»fit fphaende folidum c uius caretus,at- 

2ue axis fit AB , centrum vero fphaerae 
X&diameterDB. Fiatangulusredus 
DEF » eritq; BF tertia propordonalium » 
pofita diametro DB pro prima» & femila 
tere poligoni BE pro fecunda* Dico 
fphaerale folidum ad infcripta fibi fphac 
ram efie vt tota AF ad DB* 

Concipiatur circa iphaeram cyiin- 
diiisMNOP>&perpun<^AtD»B^F»planaagantur ad axeni 
credac 

Erit ergo^per praecedentenufphaerale folidum ad infcripti 
fibilphaeram» vt fuperficics fphaeralis foiidi ad fuperficiem 
fphaeracshoceft^fumptis aequalibus , vtfuperfides cylindri 
GHIL ad fuperficiem<ylindri MNOP i hoc eft vt reda AF ad 
BDperprimamp^paitis. Qiod&c; 

Trof<fitio XVI II. 

Sl circa circulum d^fcribatur poligonum habens latera nn« 
mero imparia9& conuertatur figura drca catemm poligo» 
fiii habebitfadumfphaeralefolidumad fphaerameam rado^ 
nem quam habent quamor fimul termini nempe 9 maximus, mi« 

nimuiqi cum duobus mcdijs ; ad quatuor miQimosiTquandd r%« 

tio 



do redae GB ad GD continuau iucmin tribui termtau . 



y 




Efto circulus cuius diametcr ABii ocn- 
trum vcro G, ipfiq; circumferibattir poli- 
gonum liabcns latcra dumcro imparia» 
cuius catctus fit CB> & conuertaihr figu- 
ra circa CB ;Faftoque anralo GDP, rc- 
^o> crit ratio rcdac GB ad GD continua 
ta in tribus tcrminis GB,GD,GF; vripro 
pofimm cft , Dico folidum ad fohacram 
cflTc , vt GF, GB, finml cum GD hvs fum- 
pta , ad ipfam GB quater fumptam • 

Fiat alius angulus ADE rcdhis ; critq; 
folidum ad fphaeram per praeccdcntem , vt CE ad diametrum 
fphaerae AR hoc eft vt EG^ GD fimul,ad diamctrum iphaerac 
/funt enim aequales G C ,G D^ hoc eft vt dupla £G , & dupla 
GD*adduasdiametros,hoccftvtFG,GBcum di^ GD, ^nemdu 
ad quatuor femidiamctros G B« Quod erat demoiL &c; m imfis 

^Mddxitm dffkm^tumfuit^ ^ifitndemMs fic . Dico ipfamEG 
bis fumptam , aequalcm cflfe duabus FG, GB # 

Qupniam ob angulum redum , redangub ABE , G B F , ac- 
qualia lunt eidem quadrato BD,acquaUa erunt & inter fe; ideo- 
que larera eorum reciproca, nen^c vtABad BGfubdoplam, 
ita eric FB ad B£ fubduplanKaequales crgo funt F£, £B« & ffe$ 
rcdae GF, GE, GB . funt in proportionc Aritmctica video £ G 
bisfumptaacqualis^ritduabusFG,GB« Quod&c 

Trofogtio XIX. 

SI intra circulum defcribatur poligonum habcns lateranume 
ro imparia , & conucitatiir iigura circa catetum poligoni , 
crit fphaera infcripmm fibi fphaeralc folidum , vt funt quamor 
fimul maximi tcrmini , ad maiorem reliquorum femcl ^ & mc- 
dium bis > & miaorem femd fimiptum ((yasdo proportio CD 
- I adCE 



fi Di Sfhspdlt^ (M^ f^dirM. 



T 



7*r9pofith XXli. 



►%-^^ 



7, 



Sl eidem/phxrae duo jfolkli {larilateta , & &mlia»,ciu-ca<pie 
idiagonalem rcuoiuKa , ftheram dtcumTcnbatur, altenun 
. verd inlcribatur; fuperficies fphsEntinediapr(^RioaaU$cn^ 
im& ibperficies duorum folidorum ♦ 

#-- ■ , • ' 

Sii circuius, cuius diamcte* A B 
aqut' ipfi duo polig<»a > ah!M%im 
circumfcfibatur, afeerumverdin- 
fcribatur , habeatm vtrlimqi latera 
numero paria > & (it numerus late- . 
rumvnius oequalis numero laterfL 
akerius, vt fph^ralia folida fimiiia ^ 
cuadant. Tumconueitaturfigura 
circa diagonalem CD • 

Dicofiiperficiemfada^ fpha^ae 
mediam proportionalem ^flfe itt- 
ter ii^erficies fadorum folidoru . 
DucaturexcentroGre<aa GLad. .., 

contadusM&Lj&radioGM fiatfph^ralMH-. 

lam fupcrficies folidi AF ad fuperficiem fphaera? IM intra ijK 
fum infcr4>V ^A ^^ folidum AF ad fphan-am IM > per 5. huius » 
nemp^vt a»i3 JS^ ad GM> per iS. huius, hoc eft vt re^^gufum 
AGM ad duadraium GM ; Superficies vero fphaera?IMad fu- 
peij^ciem Maeral ^FeHfl vt quadratum GM ad quadtiaeimi G 
A . \ Ergo ^x seqiidiuperficies foildi AF ad fuperficiem f^^ianras^ 
ALerityt j^ei^xifgtuum AGM ad quadratum G A » netnpe vt re* 
aaMGadGA.velvtre<aaLGadGC. Sedfaperfidesfphai- 
Va? ALF a^fuperficiem folidi GE^fl vt LGad GC • (quo^ fc^ 
bat6r eddeite mfo^ vtfa^auiti fftii^lupra ) crgoin C€mtmiia<prO'^- 
.portibnt^ iiint ftLperfities vniuer£i fotidi AMF , fopdrfici^ ^tw ^ 
rffALF.&fuperficiesfolidiCE. <22Pderat&c. 




CmU 



Littr Seauubis i 



*f 



Hhe fdtet itidmtiM^dfi eidtmfoUd&ffb^dlifMtdteto ftr-^ 
04 didgondkm rnoiUfto dMdffbdtd , dtirdcircMmfiriidfttrydln^ 
Pdnj^eroitafmbMtm^tsfrtftrficits in coatitmdfroforti^Mt crMMt 
wterfc. 

Vroptfim XX III. 

s • 

SPhao^alia foli4a parilatera ciicadiagonaletn reuoluta, & ei 
dem fphxrat> vcl ax|ualibus fphaeri^ circumfcripta, ioter jfe 
funtvtaxes. 

Sint ckca circufaim cuius centriHn A 6xki 
poligona diflimiia > quorum latera numero 
paria fint, &conuertantur circa diagona* 
lem« Sitq; alterius fa^onim iblidorum B 
FC>axis RCjalterius vcro nempe DGE^ 
efto axis DE > Dico foltduniBFC» adrfo-- 
lidura DGE eCfc vt BC ad DB ; 

Hocantempatet. Quoniam folidumB 
FC ad fphan-am eft vt BC ad diametrum HI; fphaera vero ad al- ^ *«hm4 
tcrum folidum D G E eft vt;diamcttr HI afl axem DE , erit cx ^' ^ 
aequo,folidumBFC*adfoiidumDGE,vtBCadDE. Qupd 
cttt&c 

m 

ff1u»Mn , 4d tMtfurn qtu/utidiuk D0£fi(f^Af tmdcM ffht-^ 
rgmiiffftvtiH^AdHH.. ... 

Citm tnimfotidtanBrCjadffkttf^mftv^ tJjdJtiyOiit di-^ 6Jmut, 
Hidend* exceffus SFC Adffharam , vt BH ddHA . Eadem 94- 

tiM€^haf^4dtxc^jfimJD6M€mtMlS\djt^JfiI)ingoex£att4, 
txceffus BFC ^dtxceffum DC£ yffifra/fh^ram erit vtiU^^d 
»D. St«d&d^ ''■/\'f 




** i ' t « \ 



•( 



fn~ 



^6 De Sfbdra, tf foliSs ffbitralih 

"PropofiM XXIV. 

, - « k 

SOIkla fph^atf a parilatera , eidem, vel asquaHBiisvfphaerisi 
lAfcripta » & drca diagonalem retk>]h]taKiuttih«er ie isbdxhK 
plicata ratione catetorum . - . / i 

Infcribantur incinculo ciittis diame^ * 
ter AC duo femipoligona ABQ ADQ 
& cbnuertarar figura circa diagonalem . 
ACr vt defcribantur duo folida fphasra-» . B| 
lia vt imperatum eft . 

Dico folidum fphaerale fa<£):um ex po 
ligono^ABC , ad lolidnm fphasrale hA, 
dum ex poligono ADC> effevcquadra 
tuq[icateti I E 9 ad quadrarum cateti IH • 
7Jmiut, SoUdum enim ex ABC ad fpha?ram> eft vt quadratum li^ad 
fri^^' nuadratumlC 5 fphasra aucem adfolidum AD C, eft vtqua- 
' aratum ICadquadratumm> ergpex aKjuo iblidum ABCad 
folidum ADC eri^ vt quadramm Ifi ad quadratum IH • Qqod 
erat&c. 




V 



Vrofofitio XXV. 

SI intra aequales, vel eandem fphasram, cuius diamcter AQ^ 
defcripta ftierifitdupiolida^fpbazralia parilatera, quomm 
duo latera fint B C, BD ;^exnittahturque ex puni^s Q D , per- 
pendiculares CE, DF ad diametrum > erit foiidum cuius latus / 
BC, ad folidom ciliusiatus BD; vt Afiad AF .:'- 

Ducantur enun oc <%ntroIadlatera£C » BD |)eipendiculjb- . 
resICIH. ■ • ' ■.....; , /■><•-■ • 

Reda E A ad reAam AB,e():viquadratum AQadqttadrA«*\ 
tum AB ( ob angulum in femicirculo re<5ium ACB^ reda autem 
BAadAFicilvtquadratumAB, adquadratumAD, ergo ex 

aequo 



;ri 







■-■■ tJter Seamdm • 

«quo re&a £A ad re&am AFtC ift vt qua^ 
dranmi AC ad quadratum A D , hoc eft 
vt qnadratuita IG s4 quadiratum IH« hoe 
6ft vt foHdttarcmiKsladrs eft BC ad foli- 
dumcums latiis eft BD. •Qupd erat 

Trofcfitio XXVL 

'• • • 

SI intra fphao-am cuius diamctcr AB dcfcriptum fit folidum 
fphaarale pi^ikteriisi /jk circa diagonalem reuolutumj 
dcmittaturque ab extrcmitatc lateris BC quod diametrum 
cbnriiTgi^ re^Sa CD perpefidicuJaris ad dia^tmm<ijfculi AB> 
crit conus cuius bafis circuhas A F- 
CBE cflttttMo vero ft AD , fubduphis 
fblidi fpfa^alis ;comis vcr6i cuius ca- 
dcm fit bafei & altitidb DB , crit fub- , 
iiuplus di£fercmia?,quaeimer^haeratii.j 
& folidum Iphacrale cft« 

Sphafflracnimadinicriptumfolidum — i.bmm. 

cft vt quadratum diainetri ad qyadratum »cateti AC (cft enim 
AC ob angulum reftum ACBtSequaHs«cat«o poligoni,) hoc ci 
vt BA rcda ad rcdjfcm AD • : 

lamquiac()nu$f,cuius bafis AFCBE altitudo vcrofitABj 
aequalis cft ha?niifphaa"io in cadein bali conftituto ; crit di<5tus, l^i^'* 
conus,hoceft hemiff haeriiiih,adcoht;mcuiUs bafis cadem A 
FC B£, aititudo vero AD, vt AB ad A D. Sed hcmifph?fium 
eiiam ad femifolidum eft vt AB ad AD i vt oftendimus fupra* 
Proptcrea conus cuius bafis circuius A f QB E, altitudo autcm 
AD ,»critaequalis fcmilbiidofphagxali^fiue fu^ 
ndis. Quod&c. 

d$ni€ro DSy fitidefkm e^c^ce/asMns , qHQfphwafilidum 
fnfierat.. 

Sjche^ 







40. 



* ^fu* defcribunttir imreMolHtione fij^r^ iihiUntism^tiSy^^U^ 
vnum en FC , &foUdumffhdr4lecir(tttr4Ml^j s<li*!(tlt^tp ft>i^ 
guUsffh^oidibus ^ efuarumvniufiuiufqi maximus tirculusjt 
tirc^ diametrum FC . Axisvero aq^uoHsfttfortionirtSta ex AB. 
^juaintercipiturinttrduasfn^neUiulAfesadipfm AB dttifas 
*xfuntiis F<^C. (jrfc de reUquis , Sadhoi Mibt • 

Troprfim XXVII. ^ '■ 

SI dacmcirciao^duopoiigooa pariUtera alterum circtun- 
fcribatur, alcerum vet6 infcriba«ir ; & conuertatiK circumi 
fcriptumquidemckca cafetum, .ittfcfj^mim va-qcir<;a diaap^ 
nalem j erit differentia inter circumfaipwm ik. fph^aqi, ad di^. 
ferentiaminterfpl«enim&infcriptum,^rtqMdrawpaIateri$arr 
<umfaipti ad duplum quadraii kterisinicriptio 

Efto circuli diamctcr AB , latus 

vero poligofti circumfcripti CD* & 

infcriptiAfi. Dko exc^ifuni, quo 

inaiu5 iblidum fphasram fuperac,4d 

cxcefTum, quo fphaera fuperat minus 

cfTe vt<}uadratumC D ad 4uo qua^ 

drataexAEw 
j^, Solidum enim circumfcripium eft 

ad fphasram vt duo quadrataCI » I A 

ad duplum quadcati cx I A ; ergo diuidendo» crit exceflfus folicfi 
^^- fuprafphaeram, adipfamfphg^am, vtquadratum C A ad du^ 

pium quadrati ex IA>(iue vt quadr.CD^ad duplu quadn ex; AB 

Sphaera autem ad exceflfum, quo ipfa fuperat minus ioIidum»eft 

vtquadratli A&adquadratumAEi vclvt duoquadnita ex A B 

4id duo quadrata^ex A£. Proptcreaex a>quo exceflus folidima^ 

ioris fupra fph^ram i ad exceiTum fph^cx fupra minus folidum # 

cric • 





tiher Secundia i ri 

ciitvcquadtatiimexCDadduoquadfataeKAB< Qgpd&c; 

Trofiftth XXVlIt. 

« 

QVodlibet fphaerale foiidum circa diagonalc reuolaeum 
(cuius latera numerp quidem paria (inc, fed nullo 
i^^ mododquatemariomenfurentur^vtfimttf. I0.I4. 
a 8- 2 2. &cO iAfcriptifibirombi folidi dupliAn eft« 

^ Sitfolidumqualedidum eft AB 
CDEFG.circa axc (iuediagonalem 
DI reuplutum . Manifcft u eft quod 
diio latera oppofita BL.FM.coijtin- 
gentfphan-am in extremitatibas At 
G, diametri AG, quae quidemper- 
pendicuiaris fit ad DI. jquadoquidf 
laterum numerus a binario tantum 
menfuramr,nonautem aquatema* 

rio . 

Infcribanmriam duo coni ; nempe ADG in femifolidc^ ha-» 
bens altimdinem HD • ; conus vero AIG in hemi^hasrio • Erit ^ c ..^ ■. 
igimr femifolidum ABCDEFG ad hemi(phaerium vt axis ad **^ 
axem , nempc vt DH ad HI , hoc eft vt conus A DG ad cottuai 
AIG (cum fint in eadem bafi \) &pennutando fenufolidum ad 
ibum co num ADG,erit vt hemifphaerium ad fuum conum AIG} 

3uare duplum erit . Propterea omne folidum, quale didum eft 
uplumeritinfcriptifibirombi folidiy QiK>d.&c« 

Si hemifpbdrmH ABCy & conms efmcmHquertCtms DMS AMr. 
Jcm dliitmdinem hdbuerint FB;erif hemiffh^rum sdfrMdi&MHi 
CMHmvtduflumbdJishefniffhfr^dibdfimeiHfdfmcem. 

SitvtfomtHriEtiHfcribdMr in bemiffhdrio cohhs AUC . 
tritcrgocomtsABC ddfonum DBEvt hdfis AC HdbdfimDEi 

K ^ fnm- 



famptif^^kniecedentium duplis ^eHf he-^ \ . u ' 
mifpharium j4B C ad conum DBE vt du^ ^^^ 

flum bafts AC ipfius hfmifpharjl ^ ad JXE 
iaftm coni. ^upd eras c^r. 

.P A 

^ TrofofHio XXIX. 



>« . -«•« » 




q: 



Vodkbet ff h§rale folidum ciaa diagonalcm feuolutuni, 
cuius latera aquaternario menlurentur, ad infcrip- 

^^ tum fibi rombum folidum , eft vt fi^rficies infcsip- 
r«,adfetoifupecfidmdrcunafcriptae. 



*• 




Sit folidum quate diaum eff ABCD- 
£• cui infcribatur femirombus,hoc e(l co 
vm ACE; ad alcimdinem ver 6 hemifph^ 
rij fit coHjis AFE^ in bafi AE.. 
fuhmt. Eritergojpsimifolidum ad^hemifpha?^ 
rium vt axis ad axem > hoc eft vt CG ad. 
GF,fiuevtconus ACE, adconum AFE 
^iuhtemm in eadeirx.bai\^,& permutanda 
. eri^f emifolidiim ad ftuun conum AC E ^ 
vt hemifph^rium ad alterum cooum. AFE , hoc eft per iemma 
pnemiiTum » vtduo circuli ex HI > ad circulum ex AE , vcl fum:- 
\a& duplis, vt quatuor circub.es. HI» ad duos circuios ex Afi; 
hoc eft vt fuperficiesinfcnpta^ intra folidum fp) israeL ^ ad femi-r 
fiipeiflciem circumfcripta^. Propterea etiam duplaL eandem 
rationem habebunt». hoc eft tomm fpharrale folidum ad infcri;> 
ptum fibixx)mbum. fbiidum erit vt didum cft... Quod &c... 

^ BoMM itiimc^nclmdhfhlidimffhafmh ffodicittiH effh adin 
fcxi^fmfthi rombum^vt infcriftus infaligono circulus adfemi^, 
circulum circumfcriftumivelvtquadr^umx:ateti GH adfemi^^ 
quadratum diagonalis GA eiufdem foligoni . 

Lemma . • 
: Si in tridnguh a^uUatcroi infcriftus fitcrtt nh^tthtt . £rit( 

circu^ 



.■*■•. 



1M 



IJierSuitnidm.^ 



) >: ^. 



'^. 



€i9€ulHsdttrcmHsdldmetafit Idtus tridnguU ^ tfiflushfmf^ 
ticircuU^ \ 



■^^ ^ «. 



'j 



E > 




A p 



, ^ Tnf&ii4iurcirculusASCiHtri4ng. dfui^ 
ktftroJiES. SitqucG funfiMmrCcntrum^ 
circuU , ^ trjidnguB ; frtftcfCH DG dufU ip^ 
j ius GC y hoc eft ifsius GA . Ergo ifuddr. DG 
^snndruflutH quddrdtiexGA. KJr^Uddr^um 
dDAtrtflumdriteiufdem GAi ^udrc etiam 
circuiMs cuius femididmeter sit JDA triflus 
erit circuU cuiusfemeUdmeter sit CA. ^od 

erdte^c. 

* > ~ " » • • 

intio XXX. 



SI clrca circulum defcriptum fueric triangulum «quflateiafn 
& reuoluatur fi^ura> erit fadhis conus aequilatcrus ad infcrt 
ptam fibi fphaeram vt 9. ad 4«^ 

£(h)circacirculum ABC trianguluni 
aequilat erum DEF > & coouertatur i^ura. 
Dico fa^him conum ^quilaterum e(ie ad 
infcriptam fpbaeram in propoctionedi^la 
fefquiquano» ncmpe vt p . ad 4.' 

Concipiatur in hemifph^io G AI conus 
GAI« Eritiamperlemmapra^cedeascir'- p 
culus cuius diameter D F triplus circuli 
cuiusdiamete;rGI;fedconusD£F adconum GAIr;uionem 
habctcomppfitam ex ratione altitudinum fi A ad AL s quse trip« 
la eft : Et ex ratione bafium > nempe circuU DF ad circiuum GI 
quae (imiliter tripla eft : quare conus D £ F ad cohum GAI ent 
vt s^ ad vnum , lumptifq; confequentium quadruplis , erit conUs 
D£F ad fphanram fibi infcripcajn» vt $. ad 4. Qupd er at &c« 




K i 



frofe- 



fn\ 



^"S 



Trofoftio XXX L 



• • 1 « 



SI drca eandeth ^haersm dcfoipti fincicdiitis» & cylifidras» 
amboaequilatdriichinttria foudat iteinpe comiSyxyli«H 
dnis> & iphaerain contiaua propordone fefquiaitera • 

Hocautempatet. t^ofitaenimfphaeravt4.eTit/^Corol« 
lariumProp. 3o.pypards/cylindrus vttf ;conus aoceni ofteo- 
fuseftinprxcedentieirevtp. Quaretriafoiidaenincintierieia 
continua proportione fefquialtera * Quod &c» 

Tr4iftifim XXXII. 

SPHan^ adinfcriptum ijblcoAum asquilatenun eft in ratioiie 
numeri32»ad9« 

Sitin drculo cuiuscentnim Am^ 
icriptnm triangulum {quilatermn C B 
D. & c5uertatur fiHura circa CH. Di* 
co fphasram effe ad fa^Sumconfi (qui* 
laterumiibiin£aiptumvt32.adp* , 

Ducatnrdiameter £F ad angulo^ 
rc^s ipfi CH t & concipiatur in he- 
mifi^iaaio conus ECF : Pun<3um A 
critcentmm mm circuli > mm etiam trianguli asquilateri 6OD9 
^propterea C H feiquialtera erit ipfius CA^ 

Sed cum etiam ICL fit trianguium a^uilaterum» erirCApo« 
tentia triplalpfius AI , ergo & circulusex CA, (iue eac A£ tri- 
))Ius erit circuli ex AI; ideoq; conus ECF^ triplus coni ICL « vi- 
delicet vt 24» ad 8« Conus autem I C Lad conum BCD ob & 
militudinem , eft vt cubus A C ad cubum CH 9 ninunim yt 8» 
ad 2 7. Qoare ex a^uo erit conus ECF ad conum B C D vt 24« 
ad 2j. Redudaquerationeadminimos terminos^ erit conus 
£CFadconttmBCDvt8*ad5. Sumptisigiturantecedentium 

quadru- 




quadniplis^bfraadinfcrilliuinilibi coauacquiUtenimnityt 
^XMikf, C^odcnt&Ci 

trofi^ XXXIIU 

ROmbnsfoUduszquilaaaus circa b^xnm defcti|itus eft 
adiplaaifptHcriunvtdiameiertjaacliuiiidlaiiBapfdenii 

Efto cjuadranm ABCOcirca circn 
Ititncuius centrum £ ; & voiuaair Hgu- 
ra ciicadlagonalem BD ; Dico rombu 
(olidumaRiuilatenim ja^um ex reucK ^ 

lutionc, euead^lixramvt diameia i 
quadratiadlanueiufdem. '■ 

Imejligatur in hemitpb^ioconus F 
GH. cuJusbafisFH.altitudo EG, 8c 
ducatwIM. 

Eritiamconus AEC culusbalid AC limiHscono FGH,vter 
gue enim re^ &; reiftangulus cft . Ergo conus ABC ad conQ 
FGH eritvtcubusBE-adcubUBi EG, nempe vtreaa BE ad£ 
L.CfunteoiniEBtEGiEI, ELincontinuaratione^ fumpds 
9utemconfequentiumduplis,eritconu$ABCadhenufphxriQ* 
vt BE ad £G, & propterea totus rombus foUdus ad totam fph^ 
tjmfibiinfcriptameritvtBEadEG, hacellvi diameter ali-^ 
(uiusquadraiiadlatHseiufdem, QiKxlJtc 

P«yoy7ria XX XIV, 

SPhzraadjnfctipiumlibicylindrumvquilacerumellvtdia» 
jQeier^uadratiad3.quart.Iatcriseiufdem. 

Delcnbaturintracirculumcuiujtcentnim AquadraiumBC 
DE, & voluanv iiguracirca cateium AG , Dico fph^am ad cy- 
lindtuinBCDE,elfevtd)aiIlctei: «licuius qujdratiad^. quart, 
hteiiseiuldcm. 

Inicl, 



' fotelligatur circa /[^maip alccr cy 
lin<irusaBquilaterusFILM. &produ^ , 
aaAMiungantur AD,<X)« Erunt 
ob fimilitudinem triangulorum, in | 
continua ration^ FA» AD, AC^^j A^^^ A:^^ rf^ 
^tquiacylindrifunt fimiles, nempc ' »- 
«quilateri , erit cylihdrus IFML ad ■ 
Cylindrum BCDE vt cubus FM ad 
cubumCDjhoceftvtcubusFDadcubumDG, fiuevt cubus 
F A ad AD , hoc eft vt reda F A ad miarta w AP.. Sumppfque 
antecedentium fubfequialtcris, ent^haera zd cyjiodru BCDB 
vt dug tert. ipfius FAad AP j hoc eft vt tqta FA ad fefquialte^ 
rainipfius AP iiiue (quod idem eft)vt FA ad 3.quar. re(Stae ADi 
Conftatergofphaeramadinfcriptuin Gki cyljndrum afquilatc-r 
rum efle vt FA ad j. quar . ipfius AD; hoc^ft vt diameter alicur 
jus quadratiad 3. quar. lateris eiufdcm . <^d.&& 




» .... 



' ' * 



Propofuh XXXV. 

SOlidum exagonale s hoc eft fpha?rale foUdum genitum ab 
exagonocircacatetura reuo4uto^ feptupkun ellconi ean- 
dem fibi bafim , & alritudinem habentis • 

* Efto exagonum aequilateram,& -^qui- 
angul.um ACDEf B &conuertatur cir- ' 
ca catemm HI ; anfcribaturq; conus AI C^ 
B . . Dico exagonale foHdum ftiftum ex 
rcuolutiotte , feptuplum elfe coni A I B • * 

Producanmr C A , FB donec concur- 
r^nt in aliquo pun^ao L, eruntque ob exa 
'Sonum , quamor triangula asquilateria O 
C A,0 AB, OBF, ABL, aequaliainter fe. 
Concipiatur ergo conus CLF perfcaus; ericque coniis AIB<IU-^ 
plus con i ALB,quandoquidem eande habet bafim AB, fedal- 

tim- 




UbtrSecuffdufl 79 

atudmem habet HI dupbm ipfius HL .^ 

latr^conus CLF . ad conum ALB, eritob fitnilimdinem, vt ^ 
cubus CL ad cubum LA , nempe vt 8. ad r ; & diuidendo fcmi-» 
folidum CABF erit ad conum ALB > vt fcptem ad vnum . Pro- 
pttreaeiiamdiiplaeandem rationem habebunt,hoccft folidu^ 
exagonale intcgrum feptuplum erit coni AIB.. Qaod erat &c. 

Trofofitio XXXV L 

SI circa circulum dcfcribatur exascmum , & reuofuaturfi- 
g^urajeirea^ciatetum 5 erit^hasra lextupla coiii, qui eandetn. 
bafim ^ &eandem altitudinem cumfolido habeat*. 



Efto circa circulum cuiufiT centrfi 

XexagonumABCDEFv&conaer- ^ , 

tatur circa catemm GH ; infcribatur 

que in fado foiido exa^onali co- g 

nus AHF, qui bafim haieaticifai.- ' 

lum circa AF, altitudinemvero QVt. . . 

• • • 

eandem cum folidb * Dico fpharranv 
fextuplam effe coni AHf •• 

Concipiantur duo alijH:om; nempe LHM ia he^fphaario»^ 
AlF f uper bafi AF conftitums ad centrum I - > \ . 

Erit crgo propter exagohum ,.trianguium AIF »|uiIateruiniK 
& ideoipTa IGuipla erk potentiaipfiusGA. CQnftac igituft 
quod circulus cuius diameter LM ^dupiaicilicecipfiu$ lG)iCH 
plus enit cifculi cuius diameter AF , & propterea conusL H M 
triplus crit coni JUP^^S^^lfg^lja dJ^tenv«iubd^-u{4S. erit coni AIF » 
&ideofexmplaconiAHF. Qupderat&ci 

Tropofiiio XXXVtL 

SI clrca civcuhim defcnbacdr lexagonHm t &^ vdltiatlff:%n-* 
racircacatett]mrencfa^tai*4bj[^uto.ad &^m fpb^an» 




r 



• • 



^o De SfUfM, (f foUMs ffhMrM. 

Eftd circa circulum cuius ccntrum I 
cxagottumABCDEF.&conucrtaturfi- ' 
guracircacatctumGH. Dico folidum 9^^ ^ 

5>haeralc i&aum , cfle ad (ph^ram vt 7* //Vx 
add. 

Concipiatur cnim in folldo conus A 
HF, vt in duabus pr?ccdcntibus propofi- 

tionibus . 

Eritcfgo(^pcr35.huius)folidumcxa- A fi F 

gottale ad conum AHF vt ^.ad vttUm,co 



m 



€ed» 



lidiun ad {phxsam vt 7« ad 6» Qupd &c 



.jqiiare 



dcfnb 



Sit eicdgMnm ABC cuius eentrum D » 
Dico diagondem ACfotentii ^Jftfcfqm 
tertiamcAtetiEF ^ B 

Hoc dutemfdtet . Ndm duStd DB. erit , -g 
ARD mdngmum dquiUterum , ob epcdgo . . . 
numidrADUtuseritfotentid/efquiter A 
fiumferpendiculdrisDEi ergo/hmptisH 
neisdufUsy etidm ACfefqmtertid eritfo 
tontU iffus EF ^ ^nodcjrc 




'.* 



Vrofoftth XXXFIiL 

SVYaaz inrcripti (ibi fbiidi exagonaiis circa diagonalemre^ 
uoluti > rerqivtertia eft * 

Sit in cif culo cuius centnim A ddlaiptiiai exagonum BCDEF 
G i iimdlifq} DH»DL » DM* DI*c<Muiercittiir figiva circa dtago* 

nalemDG» Dico fptUEraminicripti folidi exagooalis fe£|uir 

ter- 



4 

m 



«« 




lHtr Stcmubtt 

jKrdain efle • Circulus enim>cuius dia 
' meter H I» fefquicerdus eft circuli cu- 
iiisdiatneterLMCperlemma praece- 
dens ) ergo conus HDI fefquitertius 
cft conyiLDM»«fumptifqiieauadni- 
|>Iis»erit fphaera fefquitertia foudi 
gonalis* QiK)derat&c. 



M . h0€ emmfdtet txfrofofitionc 2 1. hmtu . . ' "^ " 

Trofofino XXXIX. 

SI idem exagonum duplidter reuoluatur, nempe drca caifr* 
mm,&circa.di;^oiuiem>Eritfolidumdn:a catetum re« 
uolutum , ad folidum circa diagonalembia fubduplicata ratioae 
aumerorum49. ad 48. Nempe vtraduc q. mim^p. ad nuUcem 
q» num.48. 

Efto exagoDum asquiangulum » & as. 

auilaterum ABCDEF>quod vtroq; mo- 
o concipiatur reuolutum 9 nempe circa 
catetum HI & circa diagonalemDA ; vt ^ 
inde fiant duo folida i^aeralb iater ie |^j 
diuerfa fpecie;&intra vtriiq; inteUigatur 
fphan-a infcripta. Manifeftum iam eft B 
^per lemma Propofitionis praxedcntis) 
wagonalem AD potentia fefquitertiam 
efJfecatcftiHL SiergoponamrHIratio 
nalis 5. erit AD radix ^adrata numeri 4S. ^ . 

Manendbus his • Solidumcircacatemmreuolutum» adin«' \ 
icriptam iphaer. eft vt 7. ad tf ; Sphacra autem ad folidum reuo- ^' **'** 
lunimcircadiagonalef ftytHI,ad AD, neni^. vt^.a^ lad. ^. k^mim. 
.pp. 48. Qjjiare ex aequo erit, foUdum circacdtetiim > aid folir 
.dumcirca diagonalcm vt 7. aj radicem^auadf atam dumeri 48. 

L Nem- 




$% DeSpbjtti IfffoUtsfphi^am. 

Hempe in fubduplicata radone numeroium 49» 4S* Quod 
mt&c. 

. ', • 

Sihmiffhdritm dtituMnem hiitkritfuyd0pl4m4^euiMs<$ 
m : erit himiffhdtium ad CMUmpradiifum ^ vt hdji^ ud idlm . 

HjAedth^miffhitriumABCAltitudiriem 
HBfubdufiam dltitudinis flE eoni DEF. 
i^ie^^htmif^isftmm efe vt 

eiretiUuACddeirtulurn'^^^ * '- • 

CotBciftAntur enim duo al^ coni ABC in 
hemiffhdrio , & niFfnfei^ufi DF. t!rh 
ergb eonus ABC dd eonum Z>BF y vt bdfis 
AC4dhdfimpF'yfimftifq;duflis^erithe^ 
miffhdtitmddeonumDEFvtbdfisAC A J> i* t Q 

isdbdfiml^ ^ Siii^^^ftire. 

T , . • • ■ . • 

,- - • ■ • •...,,«.. •«• 

* ■ . •■ • 

Propofifio XL.' 

SOIidum paritaterumcirca catetum reuolutum ad mfcriptum 
^ fibiconum,rationembabetquamABadBC;fadQ fcili- 
€^ angulo DBB redo;. 




H A I 



Bftopoligonum FGHILE habens la 
tfcra numero paria , delctipttDn citta cir- 
ciiiumcuiusceiitrumD. & conuertatur 
£gura circa catetum C A » fiatCj; angulus C 
DEBredu^. Dico folidum ad infcriptfi 
iibi conum F AE , efle vt AB ad BC . 
f t hmu Erit enim folidum ad jplueram vt B A. 
ad A C, fumpti^; confequentium dimi- 
dij$,eritfoliuum ad heml^baaiumvtB 
■AadDC,*fed (perlemma ^ratcedens^ • 

hcmifph3Brium eft adcoiium F Afi, vt circulns ex DCarfcnwjf. 
tem cx CE i fiuc vtieaa DC ad CB -, crg^ cx aequo eiit ^aerai. 

kfo- 







ie foUdum ad iofcriptiim fibi coniitn F AE>yt i^ ad B»C • Qupd 
crac&c 



c 



Vnfpi» XLL 

Onus inicriptus in folido drca catetum reuoluto , asqualis 
eft exceifui quo foJidumin£:riptain (ibi fphaaram fuperab 



Manente figura& conftrudlionepraecedentis . Dico fi fphae- 
ra auferatur a iolido FGHILE » quod refiduum, quod fupereft > 
ablata fphasra» acquale erit cono F A£ • 

£(l enim fph^ rale folidum ad fphaoram vt B A ad AC; & per n kmut^ 
conuerfioneii) rationis,^folidumad iilud refiduum erit vt AB ad 
BC* ^t^ ^per prxcedentem ) folidum ad infcriptum fibi conu 
eft vt AB ad BC . Aequalis eft ergo conus F A£>in folido fph^- 
ralitnfcriptus»omnibus(imuIfojidulis annularibus qua? circa 
iphaeram funt; (iue difefeqti^qua? eft incer iblidum inlcriptanv; 
queinfolidofphao-am. Quoaerat&c. 



* 



Prop€fitio LXIL 

HEmifphanrium ad exceflfum quo fua fphaerafuperatur afo- 
lidoiphaeralicircacatetumreuoluto» duplicatam ratio- 
ne m habet diametri Iphasra? ad latus pdligpni , ex cuiusf reuoliv- 
tione folidum genimm fuerat • 

Manente praecedentium figura , & conftrudione • Dico he« 
mifph^rium, ad difFcrentiaminter folidum » &rinclufam fpha?-* 
raiii^ eOe vt quadratum AC > o^ quadratum F£ • 

£ft enim iphasr a ad folidum circumfaiptum vt C A ad AB; i \ Mm 
& diuidendo , fpha?ra ad difierentiam inter fphatram & folidu , 
crit vt ACad CB ; fumptiique antecedchtium diroidijs, erit he- 
mifphaeriumadpra?di(^amdi£rerentiam,vtDCadCB, hoc eft 
vt quadramm DC ad quadratum Cfi i vel vt quadratum AC od 
quadratumFE* Quod^rat&c# 



«4 



pe Sflnara, (ffolidii jfhiralih. 



SfhfrdddfoUdumeBvtdugqmddrdtdexCDAddMfimalqaM 

13 ^^ JrdtdCDyDE. Erg^ dimulcndo erii/phsrd dd dijferentidm in^ 

ier iffam (jrfolidum vt duo quddrdtd exCD ddquddrdtum C E 

fumftifqi dntecedentium dimidijs , erit hemiffh£ri$tm dd diffe^ , 

rentidm inter/fh^dm ^foiidum i n/t quddrdtttm DC dd quddr. 

CE^fiuevtquddrdiumACddquddrdtftmFE. SlSPd^c. 

CoroUdrium. 
£onfidt etidm hemiffh^riumdd conum FAE infcriftum im 
ffhitrdli folido yejfein duflicdtdrdtione ACddFE , nemfe dxis 
eoni dddidmetrum idfis eiufdem * ^udndoquidem conus FAE 
demonjhdtus eft dqudUs differentid interfoUdumffhdTdle im^ 
/crifidmqi/ibiffhfri 



s 



Profofttio XLIII. 

I cxagono regubri (imile cxagonum infcribatur» ita vt in* 
fcripti anguli pun6a mcdia circumfcriptorum iaterum co^ 
tingant^&conuertaturfigura circa catetum maiocis exagoni» 
cirit folidum cxagonale circumfcriptum ad infcripru vi 1 4.ad 9« 
' Sitvtponitur: Conucrtaturque figu- 
rft circa AB ; circaq; A B diametrum c6* 
cipiatur fphsera , quae quidem maiori po- 
ligono imcripta erit » minori vcr6 circti- 

foipta. 
/rr 37. * £rititaq;foIidummaius ad fph^am 
•wn/ . vt ^ ad 5.pcmpc vt 14. ad 1 2; fphaera ve- 
i» ^^ f5 jid minusfoiidum crit vt 1 2« ad 9. Er- 

g6 cx sequo foiidummaius adminus ciit 

Mii/^^i$. Qioderat&c; 







/y»/i- 



U^er Secwfduf * 



8s 




TrtffoCttio XLIV. 

SOlidum ipbflrrale fadum ex reuolutione alicuius poligotii 
circa diagonalem^ad foltdum ex reuolutione eiujfdem po- 
ligonicircacatetum; eft vtrecbngulum fub diagonaii» &ca* 
teto > bis fumptum , ad duo iimui>|uadrata | cjuorum aiterum ex 
diagonali fit , alterum autem ex cateto • 

Bfto poligonum regulare quodcum- 

3ue>haDen5 laterannmeroparia> cuius . 
iiagonalis iit AB , catetus vero C D. £t 
concipiatur poligonum conuerti. duplici 
axe;nempe primum circadiagonalem 
AB ; & iterum circa catetum CD • Dico 
folidum ex diagonali ad folithun ex ca- 
teto effe , vt redangulum BED bis fum- 
ptum , ad quadrata ex B£» & ex £D : fi* 
Ue vt eorum quadrupla • 

Fiat angulus £ B H redus , feceturqf biiariam DH in I; eritq# 
£1 media Aritmetica inter £D}EH : lam foiidum ex diagona- 6. htnm 
li ad infcriptam fibi fphaeram eft, vt AB^ad CD ; fphaera verd ad 
foiidum ex cateto, eft vt CD, ad CH; crg6 ex SBquo foiidum ex i » Mm 
diagon.ad folidum ex cateto,erit ut AB ad GH,fiue utEB ad EI, 
(funt enim fcmiflfes re^arum AB, CH . ) Cum autem BE me- 
dia Geometrica fitinter HE , ED ; ipfa uero EI mcdia Aritme- 
tica sit inter eaid. erit folidum exdiagonaliadfQlidmnex ca-. 
teto ut media Geomer.ad mediam Aritinct.incer rc£txs HE,£D 
SedratiofefteHBad £D,ead.eft acquadr. B£ ad quadr.ED; 
propterea erit folidum ex diagonaii ad folidum ex cateto» itt ipa 
tium medium proportionaie Geometricuru ad fpatium medium^ 
Aritmeticum mter quadrata B£ , BD • Spatium autem niediii 
Geometricum inter quadrata BiB , £D « eft redtangulum B EDs 
medium ver6 Aritmeticum eft quadratii £D, cum femiiTe qua- 
drati DB* £rgo folidumcs diagonali ad iolidum ex cateto erit 



vtredangulum B£D ; ad quadramm £D cum femilTe quadrati 
DB > Vcl ( fumptis duplJs^ v t re<^nguIucQ BED , bis fumpmm t 
ad quadramm £D bis , cum integro quadrato DB • Siue vt re- 
dangulqmBED bisfumptum^ adquadr^ta B^s £f)« Qnpf^ 
eratocc. . v . 

AffkmffimnsfeCldngMlum BED^ medium fHfefti$nM^ efic 
i^tet quAcbtdtd BE ^ ED . Hoc enimfAHt in fHif^^^t p»kttp^ 
cunquereciis duabus Uneis . 

Ajfumffimus etidm quddrdtum ED cumfemtffe quddrdti DB^ 
e/e medium Aritmeticum inter quAdfsUdi BMy EP • ^jgfdfdfet 
quddrdtum enim BEfuperdt quddrdttm ED quddf4t^ )^lf. 

« 

Cordldrium^. 

Hicfro C^oUdriddem^nfirdripoteJi^foU^^^xdidigondlifa 
Rum,femfer minus ejfefolido , quodjit ex c^dseto ; qudndo idtmi 
fohgonumcoffuertdturcircddfdgotodlem^ (jf^cb^cdcdjiietttnk. Qty 
monjhdtut hoc modo • 

Supnidm re^dngulum BBJ>, Ifisjtfvoffftm^ ff^^s ejt^ 4^olru% 
quddrdtis BEy ED ( funt enim in continud rdtione quddrdtum 
^Bjrelid^ulumDEB^cftquddr^ftitmED^dfifi^qi dufld mediu, 
mnor eji dudbus extremis mid^nitudinft^usr.) Et efivt reHdn^ 
gulum BED bis fumftum ddquddr-. BEy E Djfmsd^ itJifolidum 
cpc didgondli ddfolidum ex c^tcto ; EritfoUdtfuo ex didgondli^ 
OfMtus qudmfoUdum^ ex ^dteto^ ^odefdt ^c. 

. SiquisdUtemqudrdty qtfo^^xctjfkfolidttm ex; cdte^o-Juferet 
plidum ex didgondU . Hocmod^iUumfrofortionenotumhdhe'^ 

; J^dfidt vt dup quddrdtd BB » EDjmuJL^ f^imffdrdtum qutd 
^exdiffirtntidrc&drumBE^EDyitfltndftufolidum 
ftkdbebit excefum quo mdiusfolidumfuftrdt mi/ius 



Prof0^ 



p ^ • 



tMriteUndm^ 



t? 



Trofojtm XLV. 

SI intra paligomiin regu&rc piuilatermn inlcribltur fimile 
poligonum,itavtanguliinfcripti bifediones laterumctiv 
cumfcripdcontkigantsconuertanircs figura circacatetum m4io- 
ris pbligoni; Erit maius folidum fphf wle ad minUs» vt funt duo 
iimul quadrata duarum diagonalium ^ ad duo quadrata minoris 
cateti« 

^iflopbligonum p^rilatemm ABC^&a 
intra quod infcrib^ftur fimilepdligoAum 
ATC&cvtidi^umeft. Conuertaturqs 
iigura circa AC catetum inaioris p(»lk(a 
ni . Dico folidum fphf rdle A B Cad lo- 
lidum A I C effe vtduo qoadrata fidiul 
chlaramdiagotiaiiamt netnpe BD> DC • 
^d duoqciadratamin^scatetiDI. Gx^ 
cttm(crib&tur folido AlC iua fpb^a^n^ 
alteri iblido infcripta erit# 

lamiblidum A£C ad infcriptam Yphasram, eft vt duo^ua- 
<lrataiimul BD»DCaddupiumqua<lraii D C (per 1 3^ huius.y 
Sphaera vero ad inferipMtfi foiidpnieil,vt duplumqu^ddm DC 
ad duplum quadrati DI ( per 7. huiiis ) Ergo ex asquo maius fo- 
lidum fphaerale a4 n)ii^4s ifil^i dupjSmwl.quadrata BD^DC ad- 
duplum quadrati OI • Quod crat S^c. 







?t^it& :XL¥i% 



J 



• «1. «/ f . 



Ilfdem poiitis: ii conuertatur Aguracirca diagonalem maioris 
poligomGC^Eritsaaiu^fididf^niiijdi^^ atis 

ACmaiorisfolidi«advtrainqueIimul»tieipp<^fenUcatctuin D 
C minoris, &quaft3iiifropartiQiidliumGF.;.^ 
gonalis minonsa4iei33katetum;ita£^icatet^ 
teftiaadquartam. -^ . i ::. 

£&ofoliduM^fofi<a»ftME<;:iH.<di.A>M«toafe- 

udum 




Udiun IBt) . vti di<aum eft . Duca- 
tur , DE pc rpendi(!illarts a*d GB>& 
BF ad GQeruntq; in continuapro 
pbrdbnc CG;GB,GD, GE GRob 
angulos redos. 

lam folidum maius ad iphacram 
cft vt AC ad HB Tper tf.huius;fph5 
ra autem ad foiidum mtnus eft vt 
HB ad vtramque fimul DG. GF 
( per 1 4. huiusj Qa^e cx a{q«<?fo- 
lidum maius ad minus cril vt A G 
ad vtramque fimul DG. GF . nemr ': 
pe quod propofitum itierat « 

; Cmlldrium. 
SisdnJofoUddfr^diSfddb exdgom gemtdJkifim^dtmmfirA 
turq$i0dpvJiUre£fdAC3i.J3Q.&GFmufumt. nemfeDG. 
iz.& GF. p . Ergo in hoe cdfufolidum mdius ddminms efet v$ 

ji.ddzi. 

Stiferefi nunc vtfoUdd ffhdrdlid dbfolute confiderdtd htterfe 
conferdmus , &hoc quot modisfieri fotefit : quemddnmdum im 
froemio oferis nos cfefdHurosfrom^erdmus . 

Tropofitio X LV 1 1. 

SOlidafph^ralia parilatera circa diagonalem reuoluci, ioter 
fe func vt pa^depip^da bafiquadr. cateti, aidtudinc ve« 
ro did gonali corumdem . 

Sintduofolida iphseralia parikteracsr 
ca diagonales AQ DF reuoluta . Sintq; 
HI,LV perpcdiculares adlatcra CB , PE, 
Dico folidum fphaerale A6C ad iblidum 
DEF . efle vt pai allelepipcdum bafi qua- 

drato HI ^ltKudioe oerd HC»adpanule9: 




llkp Skcmdmi If 

ep.bafiqiiiclcaMLV> akitiidyneLF 

InteUigaturvtnquecirciifii^ 7.M11/; 

Ikiam ABC ad fphaaranifilMri erit yt quadratum IH ad ottadcb* 
ttm HCiZue /'ftunpta comnmni aklmdine CH ) vt paraiktepM 
fiedum bafi qnsbdrato IH> akkudine HC , ad ci^^ 
faautemABCadrph»umD£F» eftvtcubus HC ad cubun» 
LF« AcMiaBraDEF» (vt nuper in altera oftendebamus)ad 
fbhduminuih DEF^cft vt cubus LF^adpandleiepipedom bafi 
^<ibrato LV » akkudine LF : ergo ex asquo erit folidum ABQ 
adfeiidumiipbaaraieDfiF^vtparaUickpipedu^ bafi qiliKiraK» 
HI , altitudine HC> adparalielepipedumbaii quafdcaALV^aK 
titodine LF . • Qopd erat &c. 

imtatdddtdpAiidiMfktifH^^JktMm i^mftM^dkim 

«w# wcmnJtriffAimcu 4^ 7 Ammfi§mi iotfiiiitiAfmMt^ 

'^nfofkm XLVni. 

SOIida fpban-alia parilatera circa catemm reuoluta inter fe 
\ funt, vt parallelcpijpoda baii qufidr^ diagonaIis,aItimdi- 
ne ver6 quas (it aequalis cateto , & qaarta? proportionalium , (i 

fiatvtdiagonaiisadeatetilm»itacateBtsacltenam»&itt 
adquanam* 

Sinr dno folida fphaeralia circa catetos 
j^^DreuoIota. Continueturqueratio 
A ad B in quamor terminis A,B>E,F . It6 
ratio diagonalis Cadcatemm D conti- 
nuetur inc^painor termims C,D,H>I « Di** 
co j primum^Gdidum ad fecun^mi efle vt 
pandlelepipedum bafi quadrato A» aitini. 
cfineveroB & F; ad fxirallelepipedum A3&S (^ PUI 
bafiquadrato C altitiidisie vei^ D&L 

NampdoMim£AUdumad4'l^^ («*^< 

M Abia 




9> 7)e Sphtifd > tf fiMs fphdralik 

A bis fumptamracceptaq; comtnuni bau auadrato A ; erit foii- 
dumpnmumadfphasramfuam,vtparallelepipedum bafiaua- 
di^to A 9 altimdine vero B & F fimul» ad duos cubos A • SfMue- 
ra autem prima^d iecundam fphsera eft vt duo cubi A ad duos 
cubos C • Sphaera tandem fecunda ad folidum fuum, eft vt duo 
cubiC>adparallelepipedumbafi quadrato Caltitudine ver6 
D» & I (imul (quod oftenditur vt nuper fadum eft in prima ffii^ 
ra ) ergo ex asquo primun^ folidum Iphao^ale ad fecundum > erit 
vtparallelepipedum bafi quadrato A>altitudine B& F ftmul,ad 
parallelepipedum bafi quadrato C altit udine vero D & I fimui • 
Qgpderat&Cf. 

litm c$mclHdif$teBfiffhM^t nntiftMhir imtrdiffd fUidd 
UifttiftM iuxtd Pr§fojiti0nem is.hmittSifitfedlterdin/criftAydL 
terA^erhciretimfcriftA idxtd i}i& 14^ huius • ^dmde i/erk 
termiuifrafortietns dljj euddant stfrofofitis^ vt im hdc , & im/e* 
quentibus ^fcidsjfri^oirtionemfirnfer edstdem ejfe , in quibufeun 
que tdndem terminis euenidt . 

Tropofitio IL. 

^ Olidafphaeraliaimparilatera funtinterfe vt parallelepfpe*- 
l3 ^^ > bafi quadrato perpendicularis , qux ex ceatro poligo^ 
ni ducitur in latus eiufdem,altitudine vero aequali praedi<5he per- 
pendiculari> vna cum dupla eius , qpx ex centro ad angulum po 
ligoniducitur, & cumtertia proportionalium ad duas pnedi.* 
da$# 

• • • 

Sintfolida fphan-allaimparilateiasCircacatetosB^&D. re- 

uoluta • ^onp^uctOr raxio perpendicularis B ad radium poligO:* 

ni A in tribui terminis£> A, £ • Item ratio D. ad C in tribus ter* 

ndnis D^,]^ continuata fit • Dico folidum primum ad fecun^ 

dum eflevtparallelepipedumbafiquadratoB, altimdineveri 

aequali^femd^Abis^&fitfi^elyiimdqifumpcis^adp» 

- • lepipc* 



9J 




UkrSicimdmi 

lepq>eduin baii quadr. D. ald tudi^ 
ne vero arauali D • femcl > C bis> 
& I femel umulq; fumptis . 

Concipiatur in vtroq; folido fph^ 
roli fua fph^ infcripta, eritq; foii- 
dum primum ad fphasram fuam vt 
B & fi fimul cum aiq>laipfius A ad 
quadruplam B. fumptaque commu 
ni bafi quadrato B • erit foiidum 
primum ad fphaoram fuam vt paral- 

ldiepiped& bafi quadrato B.altitudine verd B & E cum dupla Am 
adquatuorcubosB. Sphaera autem prima ad fecundam eft,, 
vt quatuor cubi B ad quamor cubos D; Sphasra tandem fecunda 
adlolidumfuumeft» vt quamorcubiD. ad parallelepipedum 
bafi quadrato D. altitidkieD &I cum dupla ipfius C Tquod 
oflenditur vt ni^r fa^um eft ) ergo ex asquo patet quod propo 
fitumftierat*&c. ^ ^ 

Trapo/ftio L* 

SOlidumfphaeraiepariljttenimcirca diagonalem reuoluc&» 
ad folidum fph^^ale pariiaterum circa catetum reuolutum» 
cft v t parallelepipedum bafi quadrafio cateti , altitudine diago- 
nali bis fumptum » ad parallelepipedum bafi quadralQ cateti fi« 
imil diagonalifque > altimdine vefd cateti • 

Sintduofolida fpha^ 
ralia > quor& alterum 
drca diagonalem A 
fit reuoiutunHalterum 
verd circafatemm O 
Dico folidum primu 
circa diagonalcin , ad 
Iblidu. fecundum cir- 



It Hiuk 





bis fumpcum, adparaUelepipediimbafias^^ii^^qu 

altituiiiii^ vero C 

f . h$uut . Intelligatur ia vtroque folido infcnpca fua fphcra. Et efit Ib-- 
lidiim pioium ad(fhxt9xn fitam > ¥t reda A ad B^fiimpCfcQ; M- 
dem^ft£<)uadrato B ; crit folidum ptimam ad fpbvram t^m^ 
vtparallelepipeduqfi bafi quadrato B tliitiiditie ver6 A ^ ad <»- 

^ ' bisn^ ) (lue vt ^ ujplum didi {^arallel^ipedi ad diMis cubos & 
Spha^ra vero pdmaadfecuniameft yVtdttQcubiBii ad^UMct^ 
bos C . Sphan-atandem fecunda a«l f6lidum 6m» «4ft^ vt-dm 
quadrata ex Q ad duo quadrataC > & 1^9 fumpWiyiet^oii i f iWM t 
aki&M|inH?t2i^, vtdu€MttA^ »lparatk1e^pediimlHiiitopi 
lji)uadrilxsG&D.^fttefMdite ^roptei«aex»^p«^ 

tet^uod pro^(kum ei9t • 



t$h$imi 



iProfoftfio Li. 



•I 



Sdidum fph^rale parilaterum circa diagonalem fHMidlmnny 
ad folidum fphaerale imparilater um efl > veparallelepipe- 
dum bafi ouadrato cateth^tividiiieidtagpnali quater fumpmm; 
ad DaralleleDiDedum bafi auadrato redse iilius aua? ex cenrro 




V 



tifliiitit«i aiditfctfyr«ii&a»< 



, t ■ . '* 



Sintduo folida fpha^ralias nempe 
primum parilaterOm circa diagonale 
A conijeffliril, ajterum vero imparili^ 
termiicircacatettim C reuoIutum.C6 
tiniictur ratio C a<l D in frit^teinunis 
QDiB^. X)ico primum fc^um-a4 
fecundum ((Te, Vt parallelepij^c^kfti 
bafiquadratoB» alcimdine A quater 
fumptum , ad parallelepipedum bafi 
qi4dMmM^^ 




^» 



,k 



ftnnliiniptis. 

Nsm ioHduni iprinMimad /phxrun /tam e& vt re Aa ^ ad Bi 
fiue &Difitac6inniuiu faais quadwo B j vt pxralldqiipeduni ba* 
fi^tadnaoBaittkQdincA* ailcttbiKnfii VelBtpatallelepipie- 
ABnpraedi^nm^iiainiiaHtianiviadcubumB qtkcer jumpnun 
^zra^uer^ ptiinuii kcMaiam eft ut quatuor aibi B ad ^iu* 
laor cubos C, Spii»ca«iflmqafi Jsanbia ad iblidiira iiiuni ^uc 
•IM fa gi^Ain 4 » ' h riiiis^itftiirpUfltoor cabiC« »d {»aQil<J«* 
pipedum baii quadrato C. alutudincMro jb(|0aHi%^ iC i5f fi 
cumdqplaD.nmuI fumpUl.\ ftlipterca ex a»^uo'patet quoa 
propoiitum erat . 

Tropofttio LIL 

SOIidum fpharale parilatenim cir^ catetum reuolutum , ad 
. foUdumfph.tmpariIatmim,efiutparallcIepipcdum bafi 
■xqualiquadradsdiagonalia&catetialtitudine cateti bis fum- 
ptum»: c centro 

ducitur > altitu- 

tlineue :ntroad 

unuma liumad 

duaspr 

Sintdi 
parilatc 
tum } alt 
conucrf 

nueturintribusterminisCD.E. Di- ^^^-^ ■ . 

coprimumfolidumadfecundum ef T I I 

fe , vt parallelepipedum bafi atquali I 1 I 

quadratisB&A,aIrimdinever6A, C X) B ^' 

bisfumptum; ad parallelepipedum 
bafi quadratu C, akitudine ver6 a»]nali C, & E, cum dupla ip- 
fiusD. 
NamfolidiunprisaimadfphxramfiumeftiVtduoquadra- i 

uB 



^4 ■Af SfhMTS, ffJiiiMsfihdralik 

taB&A>addupIumquAdrad A. nuefunipta conunntu altitti- 
dine A . vt paralleletMpedum bafi xquali quadratis B & A» alti- 
tudiQC A ad duos cubos A . Vel vt dii^um parallclepipediun 
bisfuft^)tuni»adquatu(x-cubos A.Sphaoraautemphma adic- 
cundam, eftvtquatuorcHbiAadt^atuorcubos C. Sphxra 
detvque fecunda ad iolidum fuimi eft vt qbatuor cubi C*ad pa-» 
talleiepipedumbafiquadrato Caltiaidine xqUaliC&EfCUsi 
di^la D.( ut oft^um fuit in PropoC 49» hxdm* ) Bcgo ex asquo- 
psuet quod pfopofitum fiierat . 

FJHIS, 






MOT V 

GR A VI VM 

Naturalitcr defcendentium , 

Et Proieitorum 

L 1 B K l D V O* 

4 

Inquibusingenium natune circa para- 
bolicam Sneara Ludcntis per mo- 

tumoftenditur, 

Et 'vnmerfd Troieihrum doMna *unms 

defcriptione femieirctdi, 

sifolmttir. 



V «• 






»7 



.\' 



DE MOTV GRAVIVM 

Naturditer defcendentium . 



> -r 




LIBEK PMIMFS. 

CIENTIAM demotuG.&Pr.apIu 
ribus quidem tradata^ ab vnico (quod 
ego fciam ) Galileo Gtometric^ de*. 
monilratami aggredilibet. Fateof^ 
quodilletotam hanc fegetemtamquS 
falce demefluit, nec aliud fupereft no« 

bis» nifi vt tam feduli meiToris veftigia 
fubfequentes, fpicas colIigamus,fi qu^ 
ab ipfo vel reliftfle fucrint,vel abied* : 
iin minus , Liguftra faltem , & humi nafcentes violas decerpa^ 
mus > fed fortafTe Sc ex floribus coronaoi contexemns non con- 
Cemnendam. 

Principio quaedam demomentisgrauium propcmemus^ vt 
aliqua fuppleamus > quas quodamjnodo opportuna videbantur. 
ad fcientiam • Deinde qua?dam de parabola , q\xx nobis ,ad. 
propagationem huius do^fh^inas vtilia videbuntur» Reli^ 
quum libri primi propofitiones emnt de motu accelerato ; 
illarumque ordo quo ad fieripoterit in tam diuerfis rerunu 
materi js , negledus penitus non erit» LibeOus alter deMo- 
tu proiedorum trai^bit, ampliata Galilei doiSb^ina > & de« 
monflrationibus plerumq; mutatis. Tabulascert^^ quasipfe 
fludio , ac labore compomit » omnes ex tabula .finuum a nobis 
folius tranfcriptionis moleflia >'decerptas exponemus nam hy- 
pothefis noflra, iuxta quam proie(5tiones furfum fadas contem 
plamur, apert^ indicauit nobis tabulas a Galileo elaberatas in 
ipfis finuum> ac tangendum tabuUs exprefs^ inclufas > & infer« 

.. N ^tas I 



p ^ ^De motu ^rauium defcendent. 

tas eflfe debcre . Poftremo normse cuiufdam militaris conftru- 
C^foAtxAtixh^c&^^s ,^uae cutildiuerfa fit avulgarf ndrmiycu-^ 
iiis ope vniueirfa Tl£S toroieritaria adminiftratur , certe \ & /cien- ' 
tificephyk>r6phos docebic quantum axis cuiufc]; inachina^pro- 
icicntis eleuari debeat,vti)Husia(Suspropofitae, acdetermi- 
nata^menfurae cuadat . Quin etiam omnia problemata iucun- 
da fcitu, vfu non inutilia, quae circa hanc materiam proponi pof 
funt , foluca vnico intuitu in.afpc<Slum dabit ; vt ibi fufius expli- 
cabimus . Definitiones omifimiis,& genere fcriptionis contra- 
Ao • laconicoq; vfi fumus , quia dum vniuerfam Galilei doCtri- 
nam pro fuppofitione pr^mittimus ki^ori erudito fcribere pro- 
fitemur, 

Aciftrus de Motu ndturdliter 4c€el€r4t$ Gdileusfrinciflum 

fupfonit , quod (jr iffe non ddmodum euidens futat ^ dumillud 

farum exa^ofenduli exferimento nititur comfrobare . hoc eB . 

SuppO' ^^^^"^ velocitatis eiufdciB mobilis fuper diuerfas planoru in- 

/itGa^ dinationes aquifitos, tunc eflfe aequales , cum eorumde planorii 

lilai ^l^^^^^o^^s ^equales fint ,Ex hacfetitu>ne defendet quafi^ni^ 

ntrfa ilUus do^rina de motu tum accelerato , tumfroie^OTsnn \ 

Si quis dtfrincifio dubiiet deijs qup inde cofequuntur certam 

omninofcientiam non habebit . Scio Galiteum vltimis ^ita fud 

annisfuffofitionem illam demonHrareconatum.fedquia iffitts 

4rgument4tio nmlib.de Motu edita non efifaucahac demo^ 

mentii grauium libellino^iro frafigendaduximus ; vt affore^ 

M quhd Galileifuffofitio demonfirari fotefi , (^ quidem tmme^ 

di^tb cx illa TheOnmatiquod fro demonHrato ex Mechanicis 

iffe defumit infecundafortefext^z Profofiticnis de motu 4cce^ 

lerintOyvidelicet . Moment4 grauium aqualiumfuferflanis inp^ 

qualiterinclinatis^efie interfe vt funt ferfendicula fartium 

squalium eorumdemflanorum . Verbigratia . 

Sintflana a b . c b. indquaUter inclinata^ &fumftis aqua^ 
Hbus a b , c b . ducantut ftrfendicula a CiXf / ad horiz»oniem 
bf. SuffonitGalileusfrodemonfirato^ momentuminflano a 
b • adfHomentum in flano chataeffe vt eH zt . adc f . Nos 
fMi4 ir^huiufmodi Theorema mn incidimus^ hocfrimtim atiquM 

dcmon^ 



ff 




Liher Vrimui t 

dtwtnfifMtone c^nfitmdyiinus: ftoHnus 

dd ofiendendum idquodGdliUoffincifi^ 
umfiuefetiM eft ^ accedemus • 

Prdmimmus . 

Duo grauia fimul coniunda ex fe mo« 
ueri non polTe , niii centrum commune B 
grauitatis ipforum defcendut . 

^u4udo euim duogtduiaita interfeconiunifafuerintyVi ad 
motum vnius motus etiam alterius coufequatur y erunt duo illd 
grauia tdmquam^aue vnum ex duobus compofitum^fiue id 0* 
brdfiat^fiue rrut. ic^ ^Jiue qualibet alta M<^hitnicd rdtune%^a^ 
ite dutem huiufmodinon mouebitur vnqudmy nificentrumgtdi* 
uitdtis iffius defcenddt . ^dndo vero itd confiitutum fuerit 
vtnuUo modo commune ipfius centrum grduitdtisdefcendere 
pojftt ygrduefenitusin/udpofisionequie/cesidlids enimfrufird 
moueretur i hori^pntdli ^fcilicet Utione^ qua nequdqudm deo^^ 
fumiendit. 

PROPOSITIO I. 

« 

SI in planis inaequaliter inclinatis, eandem tamen eleuatio^ 
nem habentibus, duo grauia conftituantur,qu«ittterfc 
caudem homologc rationem habcant quam habent longitudi* 
nes plan orum, grauia aTquale momentum^ habebunt • 

Sit 4 ^ • horizon , & pla na ina^quaUter 
inclinata cd\cb . Fiatvt j#r ad ^^,ita 
^rauealiquod i^. adgt^auc ^» Etgrauia 
na?c in homologis planis coilocentur, in ^ -^ 
puuiftis d^Scb. eiufdem horizontalis 
linca?. Conne<5lantureriam aliquoimaginario ftmiculo per d 
c b . du6to , adeo vt ad motum vnius motus alterius confequa- 
tur. 

DicQ grauia fic difpofita «quale momentum habere: hoc eft 

N M inea 



• \ 




qrpo Demofu£ra0mm defiendent. 

in eainqua funcpofitioneaquilibrataconquicfcerei neqi{ia:«« 
fum aut deorfum moueri # Qftendcmus enim centrum commu 
ne grauitatis eorura defcendere non poflc , fed in eadcm fem-* 
per horizontali iinea ( quantumlibet grauia moueantur ) repe<- 
riri. 

Non habeant {i pofiibile eft «quale momentum t fed altero 

^ prgponderante moueantur , & afcendat graue 4 verfus r , dc- 

icendatq; graue ^ • Aflfumpto iam quolibet pundo e > cum gra 

j^- lie 4 fueritiri r, & ^ in </, erunt lineas acy ^^.a^ualestquia 

idem fiiniculus eft , tam dci^ quam ecd. Demptoq,*comnuir 

ni ^r^ remanentaequaies 4<^bd. Ducatur ir/paraiiela ipfi 

€b ,;& connectauiur punCta € d • Kft igitur giauc ^r • ad graue 

txm% ^"^* '^ ad r*, hoc eft vt 4e ad f/ihoceft bd. ad </ihoC 

^AtMm. ^dgzdge reciproce . Eft ergo punftum g centrum grauita 

4ifuspom tis commune grauium connexorum, & eft in eadem linea hori« 

^^ zontaliinquafueratantequamgrauiamouerentur. Duoergo 

grauia iimul colligata mota funt , &, eorum commune centrum 

grauitatis non defcendit • Quod eft contra prdsmifTam asquili* 

brijlegem. 

PROPOSITIO II. 

MQmenta grauium ^ualium fuper planis ina?qualiter in-- 
clinatis , eandem tamen eleuationem habentibus , funt 
^ in reciproca ratione cum longimdinibus planorum • 
Sintplana 4bybc ina^ualiter incli- 
nata , & ad idem puni^um t . eleuata^* 
Sintq,- in eifdem planis aequalia grauia 4 ^x^-^O 

& c . Dico momentum grauis c ad mo- (^^'1^^'''^'^^ ^^{ ^ 
mentum grauis 4 , efle recij>roce, vi4b^ #^A <\ 

^dbc* Fiatvt ^^,ad ^r,'ita graue 4 
ad graue aliud d, & ponatur d. in plano 
/ r • Erg6 per praecedcntem erunt ipforum 4,8cd. momcnta 
asqualia . 

Momentum autem c . ad momenmm d. eft vt moles ad mo-^ 
iem ( quia funt in eodero plano^ hoc eft vt moles 4 , ad molem 

d: hoc 




Vilioceftvt dBtZd ^r. Eftci^omomcntam /-«ad 4f,vclad 
momentum 4. ipfi moinema ^* iequalc ,' vc 4 ^ , ad ^ <• . Qu6d 
erat&c. 

IdemqModBic demnfirduimtts exFrimd Pr»p»Jiti»HeyqUit 
fumfto frincifio deditcebdt 4d imfo^hiUt oSiendetitretiam^h'^ 
folMti, & djftrmdtiite ex if^s Mechd»icdfrijtcfpi/s . 

Momentilm totale grauis ad momentum quod habetin pla-i 
no inclinato, eft vt longitudoipfiusplani indinatiad perpcnd! 
culum* 

Sip^ifCA centYum z.ffhdrd grdftis in 
fUno cleudto b c . &fi$fldniferf€ndicm- 
l$mc e . dico momentnm totdle granis a 
dd momentnm pecnUdre qnod hdbet in 
fldnohcyejfevthc.ddct^ 

Producdtnrreiid d a .percontdifum^^ 
^fer centrum a . qud ideo ferfendicuU* 
ris erit adfldnum b c . ^ quolibet centro 
f. fidnt quddrantis fortiones d g , a h • (^ ducdturi g . hori- 
xontdis , c^ d i , a 1 . ^trdq^ dd horinzontem ferfendiculdris , 

Idm Angulus^ f d c • reStus eft \ & an^uU b , c .fimul ^qudles 
funtreCto , ergo dbldtislAc^ dce, dTternis farallelarum yre^ 
manent aqudles f d i , c^ b . Sunt ideo fimilia duo triangulart- 
cfdnguldiA i , b c e . Idmfic. 

Sed qudndogrdue circumferturafemididmetro f h , fiueiz^ 
mdnente funSio f tunc momentum totale eius ^hoceB^ momen* 
tumquodhabetinfituhy ddmomentumquodhabet in fitu a, 
tft vthi. ddi\ ,fiue ai.dd ii, hocejfdi. adii.velhc. 
ddcc^ obfimi/itudinem/riangulorum . ^oderat fjrc. 

^upddutem idem momentum fit grauis conftituti , fiue i» 
funUo a quddrdntis a h ,fiu^ infunSio d . quadrantis do.fi. 

uein 





.Tr% 



10« Dem0(HCramumd(fie»denf. 

d^quider» ^ftgulus fontfftgenti* iffQlinoiiiftem MnMfntffl»,tte',. 
queAuget. 

HincPropofitiofecundakerum. • i- 

Momenta grauium aequalium fupej: planis inaequahter mcima- 
disfuminreciprocarauone^cum loBigitu^inibus pianorum, 

• « 



« I 




U$mentum in ^.AdmdLe.fnomtntmf . 
f€YfY4^cedeus lemma^ ejivi cdi.adch^ 
totdeAutem momentu ddmomentum in ^ 
^.ejivtfc.ddcd; eYgoferpeYturhatdm ^ 
tationem^momtrJum^4^momentum^r . 
ejl YecifYoce \vt ci. ad c b &c. ^od erat &c. 

CoYoUariuni. 
Hinc colligitur momentu fpha^rae grauis fuper diuerfas pla- 
norum eleuationes femper effe vt linea illa horizontalii qu« a 
contadu in ipfa fphsera ducitur . 

Sitffh^Ya gYauiscsYca centYum a^W , 
flano bc vtcunqueinclinaio ^drducOr 
tUYhd. hoYt\ontalis a contalim ojlen- ^ — >v yj ^ 

demus momentumffh^f in fitu in q^uo f 

eH , ejfe lineam b d , {fofitafemfer dia^ \ ^^ 

metYOfYO momento mdXimo^fiue totalky ^^^^'^^-y^ 

:PYoducatUY hoYizon dh£y demittatur . 

^uefeYfendiculum cf.a jjuoliiet fun^ 
ifOy&iungatUYcd. 

Angulus c aqualis efi angulo Ahh. fer jz. teYt§ i eldem 
dhh. ^fi aqualis c b f . adverticem^ergo aquales funt intcY fe 
. dnguli e , d* c b iyfunt infufeY triangul^ c d b . b c freifangu^ 
la , eYgo.fimiliafunt intCYfe . Sed iam oHen^imus mnientum 
totaleffhaYa admomentu quod habet in eleuatoflano effevthc 
ad c fy nemfejvtsf/adiameter e b • adhori&Qntalem b d , qu^t 
intYaffh^Yam a contaStu ducitur . 

Siwrograue nonfit ffhirayfedqufdcunqifoUdum z.habe-^ 

bimus 




465 




Lther Prhmtsl \ 

himtifnihihmnus fmguUeius momen^ 
td infingulis flanorumeteudtioHibusfa . 
tilime . Solu€mus.€tiam ProbUmdPap* 
pilib.S.Profof.p.famofum afudGuido^ 
baldum^ & Cabeum (jr^. 

Sitgraue a . inflano a b , c^ qusra- 
tur in hocjitu momentum eius\ fiuefotentia , a qua in hocplano 
^h .fufiinetur . 

Ponaturmomentumtotategrauis ; vet^potentiam quafufii^ 
netpondus a . inptano perpendiculari efie b c , cf circa b c ere^ 
Clam ad horiicontemfiat femicircutus c d b . quifecet ab . in d. 

Dico momentum grauis a ^fiue potentiam qua iltud fuSlinet 
inplano a b . efie b d . Ducaturperpendicularis cf.a quotibet 
puniio e. & erunt trianguta c b d, b e i.fimitia ; quiacumfint 
njtraq\ reclangula , anguli etiam c b d . c^ ^funt alterni . lam 
quiamomentumtotale grauis admommtum quod habet inpla^ 
no eb.efivt cb.ad c(y erit etiam vt cb ^ ad bc\. ob fimiii^ 
tudinemtriangulorum y Efi igitur momentum grauis inptano 
a b • vtlineaintercepta b d . ( pofitafemperdiametropropotali. 
momento.) 

^o adpropofitionem Pappi . manifefium efi , fipotentia b c 
dquattir totali momcnto b c . potentiam b d . £quari momento^ 
inptano b d .^arepotentia b d fufiinebitpondus a .inpropofis^ 
plano bec^r. ' ^ 



.f ', 



Scholium: 



lamdemonfirariprimumpouB Pro^ 
pofitiofheia Gatilei de motu accelerato . 
Sit enim ^ngulus a b c redtus^^ zc;per^ 
penciictilarisadhori\ontem cd. (jrpro-^ 
ducatur a b A.Erunt d a^ac^a bycwwmf 
proportionates\atper2.huius^motMnt}i^ . • . ^:. 
in2iC.admom^ntdinzA.efireciprooe^td,d^diC.hoc efivt 2iCy 

ad ^b^ Ergo tfi homotogc^ momentumin a c ^admomentum i» 

a b ^ 





ijpr^ De mofugrauium aej 

a b , vtfpdtifim a c , Adf^mum ai) ; quare eodem temftmferd^ 
gentur tpfaffatid a c &z b,- Sufponimus hlc cum iffo Galtleo^ 
^ebcitdtes in diuerjis flanwum imdiuatiombus ^ ita ejfevt 
funtsnomentaquando eademfuerit moles. Sed cum angulus 
a b c.fonaturTcSius^erunt b c • a b />? cir^ulo cuiusfublimefun 
Hum eff 2L,^ diameter ac . ^jipdiir^. . 



PROPOSITIO III. 



M 




Affumi* 

turdGa ^ 

uum 6. partium^qualiumeorumdeniplanorum. 

^^^V ^ "• ^^9 ^^^ 

accd. t r 1 

Sintparces a?quales aby 4r»pIaaorumincqualitrrincIina- 
toruni, & eorumdem perpendicula fint bd.ce. Dico momeo- 
tum grauis in plano ba^zd momentum eiufdem in plano c a^oi 
fcvt bd y^d te. 

Ducatur bf. ipfi c ^ • ^quidiftans . £ric« . . 

queperfecundamhuiusmomenmmin^ >i^ 

a 1 ad momentum in bf hoc efl in ac. 
( funtenimptana^/i ^r parallela)vc 
fb^zdba^ hoc eft vifby zdca (cwm fint 
Muaies partes ba^cA) vel vt ^ ^, ad ^ e. 
( luncenim ^uiangula triangula/iJ d^ d 
f r • ob lineas parallelas • ) Ergo momentum in ^ ^, ad mo-^ 
mcncum in r 4 , eft v t ^ ^, ad <-/ . Quod erat &c. 

CoroUarium . 
• Hinc manifejlum efi momentdgrduiumfuferfldnis inaqud^ 
Uth incUndtis effe vtfuntfnusreffi dngulorum eleudtionis . 

Sludndo ver)>ffhffd ttonmuedtdt in dliquofldno liberAfed 
dUigata adextremumfenddidmetri^ manente alio extremP^ ipfd 
fer quadrantem circun^efdtur^ erunt mometa eius vtfuntfintts 
fomflemttoti^^mguhfum elcuationis. Ndm momentum^ a , dd 

momen^ 



405 




.v; • ;. H^ Trimm. 



AUtcrbMiiUmMs menjkf4m momenU^ 
ffmnffhfrf^ femididmetro' ciYcumduU^^ 
in^noquq^l qHAdrAntisfHniie . pMende^ 
mnsenim\ 

MemenidffhdrfferqHddrdntemcircHmdHBf epvtJkntH- 
ne^ herizsontdUs , qnd dfHnSfo CMnexionis /jfh^trg cnm didme^ 
troyintrd JfhdTdm dtKHntnr ^ 

SitqHddrdntiscemrHm Zy/jfhdri 
^duis circd b , dr dfnn^o connexio^ 
nis c. dHCdtnr horizontklis cd . 2)/- 
€0 momentHtn ffhfrf ejp cA. (fojitd 
femferSdmetroffbfrn fro nuximo /?- 
mfttotdli momento .) jDemitSdSHrfer^ 
fendkHlum bc^ MomentHfn totHm 
ffhdTd ddmomentttm qnod hdbet i9i\j . 
. dSlvt ^fj^ ac yvelvt ba, ddzt^ 
boceSf hc.dd cAy&fHmftisdHfUs^ vt hc/ didmettr dd cd 
^orizontdlem inffhdrd , jqHddHcitHr ijnnifoconnexioniu 
^oderdt^c. 

Si v^^ grdne chrcHmdH^Hm nonjtt 
Jfbnrd dabHntHr nihiiominns fingHld 
tins momontd-^ocmpdo . -^Hfnftoin ho^ 
rizontdli Umd qnoUbet 'interHdtU a b • 
^dt circHlHs z^. 2)i€0 circHiHmhHnC 
fngHldJingHldrHmcUudiumHm snomen-^ 
td metiri: Et momdntHmgrdiHsin c. -ef 
fe Uncdm inttrc^ftdm ad . (fojitdftm^ 
ferdidmetro a%.frotHdUnfomento) £fi enimtnmentHmto^ 
tdle ddmomeiftHminc.vlt e a .ii d^i^oteBvtc Zddo^f^vefvi 
^B,dd^dnemfedidmeterddUntdm^interceftdm. 
Eddem ^d^eritPotentidquffuftinetgrdHeinJ^in C^ flftf^ 

5^ . • ^'''-«i X 





< ' /t 



* k- 



tolH 



De mofi ftamum 




fonAmMfottntidmqHdiUMdfufi^inain t. ejfc Z^. 

Ajfumffimus tnAngulk acf, ^dg.efftfimiUd^fm^ tum 
fc{fdnguUfint y hdbent dngulum commune^dd a « 



\' V 



>« 



« Scholium J 

l>imcnfirdnfccundumfotcfiPr0fafitiofcxtdpdUlcidcIi$ 
ittdccclcrdtcfcr tcrtidnihuiui ^frfmijfo koc Lcmndsc % 



\ . « 







Si circd cdndcm fciidm lincdm a b .y^r « 
fitftmicircuhtSy & quddrdns^ drinquddrdn 
tc ducdtUfqudUbctftmididmetcr b c • Etit 
b d» ihtcrccftdinfcmicirculo dqudUs iffi c 
C • ferfcndieuldri in quddrantc . Ducdtur 
tnim ^d.tunctridnguldSidhybce.erunt 
^trdqirccfdnguldy&dnguU abd, hct^ funt dtterniy erg§ 
funt Squidnguld ; bdfes dutem a b > b C • funt ^qudles ^ quMrC 
ctidm bd^ce. UtcrddqudUdfunt. ^uod crdt ftobAMum ^ 

Propofirio Galilei Sexta. 
Cum iffo GdUleo Mechdnic^ dcmonfirdtd. 

Stt circulus ddhorizantcm erecf$is a c d 
b. DicotemfordUtionumfcr cb .db. 
q/fcfqudUd. 

Sunt cnimfiifer fUnis ch^dh.fcr ter^ 
tidmhuius , momentdt^t e g, fh , W efivt 
C b iud d h. fcr lcmmdfrfcedens. Ergo cum 
fintmamcntdvtUngitudines fUnorum\eo^ 
dem temforefcrcurrenturiffdfUnd c b d b > quod crdi firofofih 
$umdcmon^dre(^c. 

P R P S 1 T I O I r. 

^T^fmporalationuinexquieteperpIana eandem eleuatio^ 

jL Qem haJbenua» fuot bomologc vt loi^gitudmes pianbr&. 

~ g^ 




107 




Litar Frimus\ 

Siot pkoa dhr^f, findwt dejuotio* 
nem ^^habentia. Dicotempus latio^ 
nisper. 4 1 jultenipusper 4k efT^ vc dc 

Sitip(anu{i4^»4r terti^i pr6poi:tio« 

aalis d € . MoQientum ergo in plano ^ e 

ad ihomentum in plano db^ eft vt db ^ 

ad ^r.(perfecundamhuiuS|)hoceftvt dc. ad^ir « Quare 

lationes per ^ c , 4^ temporibus aequalibus abfoluentun quan- 

doquidcm ita funt momenta v t longitudines /pati orum • Pona* 

mus latQ tempus per d e^ eflfe mediam proportionalem dc . Er^t 

tempusper ^i^ Jpfa di ; tempusergoper dCy fiue perir . Cni 

«qualia tempora funt^ Q&dc^Scpcr di cA ipia d t &c. Quod 

erat &c. 

Aliter. 
ftfcedens T hecremd pcterdt demenftrdflfine vlld fuffofitic 
^nc. DemanfirM fnim Cdlileus in Prof. 4f. demotn accelera^ 
tOytemford Idtionnmfa chordds omnes incirculo aqudlid effe. 
idq; tribus modis frobdt .infirimo ^ & tertio fubefi frincifium 
. fuum nonfdtis euidens ; in/ecundo vero nihilfupfonitttr , fra^ 
ttr iim di^um Theoremd Mechdnicum i quodjt^ iffo teHe^ dc^ 
tkonBfdtum dniedfittfdt^ ex iffo immedidte^ tdmqudm Cttrolld^ 
rium^ necejfdrid illdtio fud tertid Profofitionis^ immo &fug 
fetitionis yderiudrifoterdt .' Std quidlffe tertidmfudm Profo^ 
sitionemy qufnobis qudrtdefi ^ medidntefudfetifionefrobdt^ 
ms iltdm dbfoluti ofiettddmsu ex frofofiti^nibus ipsius GdUlei^ 
^UftmlUm fofiuldtutn includutot . 

SiniduofUnd a b^ a c • quorttm eddem ele 
mdtiosit ad , Dicotemfus Idtionisfer ac, 
ddtemfusfer ^h^efievt ^c.dd ab .fd£fo 
enim dnguk a b e • reHo , dgdtttr circtdus chr 
cddidmetrum zt^uitrdnsibitfer hy&fro^ 
ducdtur a c f « Erit z b . medid frofirtiond^ 
lisinter fayZC^&crunttemf^raferaby af . dqudlid^ vto^ 

O a fien- 




roS^ Demfu grauium defitndent^ 

RenditOaUleMssintfUciier exiUoThemmdte Meckdmfosine- 

fudfnpfosisiQnc . 

Si iYgefonamnstenffHs Utionis fmc^iffjt iffm a c erit^me 
did fufortionalis 4 b , tmfns ftr a f hoc eft ferfeiffm a b .• 
(jrc.^dre temforM Utionum ex quietoferfUnii edndem elena^ 
tionenihAhcntiA funt homolo^e ^^tlo^giiudinesfUnor^tmf^ 
bp€ d(monStr0kimusJine ilU fetitiottf , tmus ^Grstatemfequeto 
ii Theoiemate oJiendeinMs .. ^ 

■ 

F X o F a s i T I O' ri • 

GRadus vdocitajis ciufdem mobilis fuper diuerfaSpliah<M 
' rum incttnationes acquifiti , tuucajquak^sfunt cum.ccK 
^^ "^ rumdcm planorum deuatioacs aequales fmt • , 



.j 




Sjnt d uo.plan* a h\ 4 e .inarqualitcr ihcliriata, quorum cfc*- 
U^lionesflntiqualcs, vclfiteadcm^^. Dicogradusvddci- 
tatisjicquifitosin ^.perdefcenfum 4^,&in ^.perdcicenfom 
4L^ . aqualcs intcr fe effe . 

Quicunqj enim fit grad^us rdocitatisaqui* . 
fitus in by accepto cius fubduplo, graue mo- 
'tu.«quabili > & teporc cafus currit id^m fpa- 
"tiumcafus bA. Iccrumjquicuncpfitgfadus. 
vdofitatis aqiiifitus in .« , acccptodus lub- 
duplb, eraue motu acquabjli^ & tcmpore cafus currit idcm fpa. 
'tiumcaTus<r4. 

Tempora igi tur , & fpatia fiint proportiohalia nempe. Tcms- 

pore b 4 cuiritur fpatium b 4i motu «quabiJi: tempore autem cM^ 

fif4^ di currituirfpatium C4 niotu aquabili,ergo gradus vdocitatis funt 

^•^^ . . aeguales.C^rcctiamilldrumdlipfiiBqualesertmt;& ideograi 
^^^*'* dus vdpcitatisin/^,.3rin c. funi a?qualcs ..Qued erat&c^ 

jiliterfir circutUmfiXtitProfoJitionis Gdliteifacit} demou. 
Jhdbitureademconcl^ohocmoda. ^ 

JEx Lhcormififa Mechamcv deduxtrat Gatikustcmfvrafcr^x 
Tlji^a t- . (^^^'^li^n ife., liicoxroo ylm^cfnsinfuncfis b^drc. ^r^tr 




• * . LiBtr Frimus^ l 109 

itiffm^eddem dUhiulme.^ ^ ex quieiein a ttefcemdentiumj 
4^qeiaUs efie ^ ' . 

^SfiA enim^a b ,'^. ^f • sqttdHtemfweperAf- 
gMMu^) erumfimfesns w-.byC^ i.pHmiiSiVtfunt 
f^tiM^eMf^ «ab» \i ^(Aceieptis enimearmn > 
fkbdsifi^ nBpi4&temfeare^^.metu4tt^uahili 4Uf^ 
rsentsir ffatiA b a S^quatefubdsifliilk imfetus^ 
/iftstwffatid^^freftered etidm ilkrtimduflt 
%ft eddfmffdti^irMnt»)Jj»feitts ergV Uf* ddim 
fetum i\ ifivt ab^dd 2i i\imfetusverhin {^.ddimfetmn.its^ 
G • eSivti-A.ddjib,^ (^nimfe^vt ttmfetrdf quid a b , medid: 
frofortimali^^fl^inter fa^ ac-) Er^ensqudUyimfetsts^in 
ls\i*ddimfesMmin!C^ efiis^t^^h^^dd iffdmmet^^b,^ Jludre 
imfetusinhj & c.funtttqudks.dre. Vetfic:^ 

Grddus imfetus in.cM^gr/tdtftnJtt- Jiy,€fivt €Z. dd zhy 
wl hzydd 3i£jcum/9ffj c:k9 ^htaf.fintincantinudfpet. 
fwtiorte^ Sedgrddtfietidm imfe^tts in h . dd grddum in f » 
efi ftt b a w</ a f ( vtfufrd demonfirdsumus .) £lujtre vterque 
gtddus c, &b. ddet$ndent,i.^^itn^gprdmnch^et s & idetf. 
^qudUsjunt. ^oddtf..:^ ...... 



ferpriml 
Gal. dc 



1' .t 



HtncfroCcrolldrio cxtrhaemus iaquodiniffi^ 
frfgr^demenfir^tiotfisjtJienfk^fijnemfe.Jm 
feestsfii(4uiHminfi^ecboffidtmu^.^^sdi>qMd^ ese. 

femaofuhlimidefcjUdnt^AfJfiV^^^ 

d9ifd4\bof.tfijmtfettiiii$$s^^ A$}^ 



« > \. 




\ • 



■i\v 



J 



•i\i 






. Scholium: • 

Ctfm deincefsfufurutfitfirmo^e lineis qstds fdrdholdt ve^ ^" 
cdntj nop^if^e^uenikitJi^yfdtfe^ft^ illdrumfaj^udsin or^^ 
din^mi^i^ ctn/id^rmtis^ fjfu/^ qstdtUvt: nec^ffaria ncbir 

j^rfmonfirjtre . . iS /V emmfiet vt£dratJatds dcctdcrejfof^Ktus d£ ' 

-0 l 1 Cfiikt' ' 



I ! o De motu Cr^mim deptndent. 

i tiam (quodnon fcriffit GaUleus ) nMurdUtcr cddentitmjt mt* 
ti^d^vnicefaUAm^ PrfmittinmsfrmffrtdStiqtMitJtiffofitttm 
vnictcrfjmCaUleidoSirindm de mottt c^iilius o^dm vefiigidfc^ 
quittmr^^fAttcMidijAddamT^heoremAtaaitiffc rte^eStA colU^ 
gimsts r HicfratifttpftfffomtHtm ma Prftfcjiiimo^ ie Pard^ 
boU^quAs ipfe ofafifuo dc MotH Pfoicitcrtmi^frffgts^ idterAm 
Afollon^i quidem fedMdrtcfrofrio dcmmfirdtdm AGoMUoi 
alteramvfrVfe/titHf etcJfoHonqUk^.frof. ij^^drftmftdm^ 
(^demonfirAtdmk ^' '^^^ ^- c ,i 

^ ^^f^irna^cU^uiftfmodi. 
Une«,qiJiCfftti%|>arabolaiB bafiparalfcfadueuimii:, fonc 
in fubdupla ratlonc pottidoum diataetri ad^irciticeBi parobol^ 
inierceptarumV' " .'> -- - 

Secitsidd^cfoifi.kiC\ ^ 

* SiinparaboIaaHqtiodpunduin d futna-- 
tui-exi^db iinea docatar bafi p^rallela db » 
& portioni diametri b c . ad verticem inter- 
cept^» afqiudis reda linea c d. ponatur in di^ 
re^m« Redalinea dd^ qwe ab extremo 
pofitxlineastermino i/:adpiui^m 'd in parabola fumptum ^ 
ducitur,parabolam conting^,^^i^e^ a nobis aUquo mcdo oBc* 
dctUT fofi Prof: ty\ ' * . X ) > . V .^ 

Hac iffe • Hisfrfnti^tciiqHd nol^is cfortund dcmottfird^ 
himus : Etfrimb animadHcrtcndHm -eH^HUodnjnaqHaq^ptr^o^ 
laqHandamrcffashUncdmfccHUdrcm nabct\ cmns frifrictds 
fTAcifHA hac c0. : DmSia sntra fardhlamqi(dcHnq; hnca hdfi 
f araUela , quadratHm dHiiadqHalc cfi VeitangHlo^qi^fiib Md 
fccHUari Unca ^ (^ fortione diamctri advcrticcm fordholf dh^ 
fcifiAyContinetnr. ExemfLg. ^adratHmrcUa ab fqndc 

cH re£iangulofHh b c e^ iUdfecHUari Unca contentoi (^ hocfcm 
:fcrvbictimq;fHeritdHiiazb 

Vocatur aHtemfecHUaris HldUnca LatHs Rcftttm\^ 
^HaverodHcuntur dquidifiantcs hafi ^ Ordinatim dffUcd* 

tadicHntHr,. 




■\ . 



tikryPnmJStr "* iir 




/ jf f s I tr I o r j» 

MAnente figura >&conftiiidione fia^ 
dem quam ponit Galileus ih arima 
iamdidarumpropofitionumde Parabola. 

Fiatvt^^>ad^^.ita^^.a<i 4^^Dico 
^ r • cffe iatus reifttim * 

Suiimtur enim quodliberpun^ton in pa^ 
rabolaquod (it/&ducatury^.parallelaip 
ii db I item per g. aganir ig L par^la ad 
r ^* eritq; ^^M parallelogrumum.£tquia 
fadumeft vt db^ ad bciu bhzA ^r.erit dgadgiM ab ad 
^^•hoceft vt bh ad de iiuevt^//ad At. ReAapgulum er- 
^ogd ^.a?quale erit reaangulo ig /, hoc eft quadrato>^ , Eft 
crgo r 4 . latus f edum « 

Corollariudi . * 
Hincmdnifefiifmeftyfilincai^ . eriimdtim dfflicdtdfre* 
ittcdtur vfq; ddvherieremfcmtf^dbeldm in m; ipfdm g m f* 
qudlemforeiffi fg. eciem enimi^d^ ofienditnr qHdc^tditnm^ 
Oi^dtfifnddrdtmm^fifqtfdri.reffdnguU i^letc. 

fjtoposiTio r I /. 

SVblimitas parabola? apud Galileum y quarta pars eftlate^ 
ris redi eiufdem parabola? . 
Maneatconftm^O Prdpofitiohis V.deMom proiedoru 
Galilei > qua ipfe reperit fublimitatem parabola? • 

Fadum ibi f uit vt ^ ^ • altitudo yzdbc /"a^qualem nempe & 
nidio bafis dej^xtxbc.zJA aliaito» qux fit b d^ &luec erat fub^ 
]imitasapudGdikum>pro^bimufqiip(a^cirequart^ late- 
fisreftipartem. - ^ ;^,^^ 



Quidratum < 4 . ouidruplum eft m»drati 
ic. hoc eft rei^guli di di (p&t canflru6li4L . 
aem ) ergo etiamredangulum f ub ^^ ^ , & la* . ^p^ 
ttreredoquadruplumcriiemfdcmwr^^iftd s y^"! 

communis t&4i6 altimdo redangulorum, er« j^^ 

go bafes, hoc eft latus redum» quadrpplum 
crit fufadimiutk i^d.Q^dScc 



f. 



Ikfiniclci* 
^4ndoy^tinprdct4eMifgitrd^ftfmitttrmA9^fdy,4Mf sx 
n;muclincd h i • q^tf d^dlisfit qttdttdfditti l^ttris r^ili^ ttmc 
funffumivocdtur/^CftspdrsAolf. Hd9s^cB$mf€jrgdtS\fMt^ 
ii$imfuiUfdt d > ctjccmm i ^qudliter diMdrc i vcrticc fdrdbc * 
Id i ncmfctdJttitm vtrittqiqudtttddfi qttdrtdfdfs Utcris rcHi . 

yy Edalineaqu^^ex foco parabola; ordinacim applicatuc^ 
J[\. dupla eft portionis axis ad verticem intercepts , Vcl • 
cqualis eft femifli lateris Re^^ 



Sic latus redum dhiti fixrus d^ Redaagoh; 
hiA bdd^ quadi^upium eil quadrati 4d..{s^ 
cum habeaixtcommunem altttudinem dd^ ba- 
fis ^ 4 • quadrupia eft bafis d d^) ergo eti» qUa* 
dratum cd . quadrup^Iumcfteiufdemquadra^ 
ci ^^.Eftigitur^^duplaipfius^^ifiueasqua 
lis:femi(fi lacerisrcdi* Quoderat&a 







f R f X> S 1 T I I X, 



« 



SI Parabola? quotcunq; circa eandemdumetrum fin^Iinef 
qua? in iUi$ p^ciinatim ducumur , proportionaies erunt • . 
J>it fjiametercf^^jn^fnupis ^i.ordinatimautemdui^iiat c^ 
€d^^h/\tg.DkQtSi^yt cc ^dglfpilxdc.^Afi.. 

Sunt 



Sttnteiiimquaclrata> *c»zd g^t vt tt^ 
CtxdczAdh, Q^ulrataedatA ^f ad/? 
funtvt dCt ad di, Ergoineadein ratio* 
ne funt quadrata iner fe i quare ^ ceda r r 
9Agbi\iaicStdcziifi, Q^etat&G 



PROPOSITIO X. 




* J 



TEmpora ladonum ^ qideex ^mete fm^ 
que funt imer fe vt linex :• q paraboia a^liott^ aoipa^ 
per quas grauia defeenderliar • 

Sint fpatia quadibet Ah,4tt&' 
ue perpendicidaria (iue inclinata * 
&circa diametnim ^rfiatparabo 
h quaEiiibet ^f/« .'«i^ biCliAatiiai 
ducantur ^^. r«. Dicotempus 
pcr i(^adten^usper«tf.efleyt'^ 
J,zd r^.Stfflt emn» teo^Hwa m .. 
fttbduplaratione ipatiomm ex Ga 

lileo,(ed line« ^^»«)r.iuntinfubdaplankiioiieipiKiorUfli(qata 
quadrataearufuntvt ^^,ad ^^.)ergoeademratioeft &tein 
po|U,& linearii ordinatim^ad ipatia applicaurQ.Quod^etat; ^ 
, ^ * ' ' AUter. 

Sifonmus temffu Uti§»hfn l*m nShm a£ effiiffiim^ 
mttUtmsteaimterittemfusfmki ^medidfrvftrMtuMsith 
terfftabihemfeifjrd.hd,'(^ toi^fgrzc midfMfrMt» 
ti0fkUtfce&e.(^jStJe/»gtdis, ^gdre^, « : ' ■ r 




•*.' »' • 



i ■ 



uj" . Cmbllarium; 

Bimmdtttfeftimi eB imfetus grMMim i»piefrti§mmidi4^ 
metrifdrdtilStefeittterfevtUiic», i»4mdi»4tim Hfflicsm^ 
md^^^xtrm^iffmMmfmitmimpui&M ^ S»mt eiiim ex G»ii, 
U»,i9ifftt»iVtifr»temf9r»t fed esrdi»»tim d»a»f»»t vt hft 

9ff^M,erg0itifet»sf»»tvttrdi»iitimd»0»x^c, > 



i 



Ii4 ^e moHif§¥iMtmi^endent. 

SI Line^a verticr:patab«teT%jii4.ic£tellfim ckKaii0ir« 
enint itnpetus in fiot liaatfiiiii»A^ti^l}Qr<UBalEim (Sx ij^ar. 
rum terminis applicatae. 

Sitparabolacuiusveftex 4«& ducanturex 
veitice^iKr>4f. i;)kdiiiip^KiQ:A»4(in<fit7: 1- .^ 
vtit^ad^e* Maoi&(lunu|ftt'€^knpetO)& .-7 
in ^, & r.funtijdemacin d. & r^ In^t aur 
tem&in e. funtvt ^V.ad ^.(per Corolla- 
riumpr»:edcnsjierg6eciamiai^& /#^fiiOt:vl^ 

idf^ti. Quod&c ;./.u . : ,..!. ; . ,1; 

PMOPOSIT / ft JT: /. Jtt;. 




7' OJ . i ,1 :■ , .- " 



TEtnpdra latfomtm, qnae IrBciceidobfiun^r pfHVdnes dm 
metri ex pun<5lo fuUinitkfttttcioiiecfe \itichpi?da; qpae e^ 
codempundoful>limiducuntur,adpiHi^jperiE^e9}si9vqu«^ 
ciii^Ci^vditUbtitndttdUtcxtq^^ .-.A 

/jSmi)afittirvtcUNiq};42>4'<&ducaatucoiidi : i^U^o-: 
natim tJ,fe» iunganturq; ddfd^i.Dico tem D . 
pc»»l^f Qnum.p«r .^ A » 4«* pocti^ne» diaraetirt - 
itaiii^iQjterieyt%)t'.4-4«E>Ar.. x -,■-'- '.vs^-. 

.^^enioApQBattir.tempuApeiv ^ettciAfak \:>^^ 
tempus per 4^ ip£a ^dusois^ fi&jne^ fiKQpot 
tionalis. Etcumtempusper 4/fit 4/,eritte' 
pus per 4 r . ipla de mediaifvt^htaililis « Qgare Scc, ^ 




•j - 



• N • ^ • 1 



i > ^ • . *?^: 






:a 



iiintinterfc* .. ....,.'■ <.... •> ^ . ^. ..-^isu ;>'••<•, -.^^.-sviv 



• ■ «. 



t.r 



>f«i 



^ tMmmt^pif^ tmttmmim ftut* fHffi- 

circMlus a b c , CMius funiiitin pikme fit a . & 
-chord^t a b , b c . infemicinMl$Mid€mfMnBitm 
h^ftiftatf . Dt{97e&if^ /)« x- h % rm^ *tt 
cfuiiteiH A,&pi^'bc'i»fii/Miepei0'b .e/ei0fii4 
lia., VtrBmqyeiirimnft^MiSfi^,^^y&fer bc 
Mquute e^ tempeti fer di MM mt rt t m zoexGMilee 
ergdfuntiquAliMhit9tftttmf9rMf«rtrttU»*4f ac, ab»bQ/ 

^id^c. \ ...... .-. i 




•t ' 



^fi» « F 9 Sil T / O X •/ /' /. 

T<nnp<«r«lBaomimp«rfeciiieBaiiactr<>s ijaaJr^nrisercfti *• 
qualia.ftmtceoi]sanbiis rtim iccaiuiiim , ,tiim«tiaHi tangeotum 
aagulonimooi^fefQcdtt elendonis, qua tomt cietiatc <u- 



>) 



« « 



Sit quadrans ere^his dic. femidiaineter» 
qua^cuna; bd. Sc tangens 4?ir4 ^ofoJt angulum 
t id^Y^rettuoiefiicfcft* Oico tempus per i/^f ^ 
<|ualeeircteaiporiper fecantem ti^ (iue per 
ttfRgemcn t i^« quairum vxFaq(« beoqpe ^i eft 
fccans » e d tangens anguli Ahdy nempd GOtti 
plementi eleuationis femidiamecri dh. 

PropoiikUmautctflm^^fkimeftpth^hnii^ cft 

criad£ulum.^^^. litre^ngulttmL^ &ideo ten^oriipervi^m 




P R Or^ O S I t I & X t y. 



n 



S/ ^Mb oliqMo pMt^flAitd^UMerfiiittil^ititUitmttdig^idijm* 
tMti&MbHlt e0daitfmtatdiittittM»tMtff$aii4etdeMtem 
ftttfimtd i tfitndittmtiUMetti Ctt^^ntitPnfuf.M, dtmttM Mt 

teltrMtygrdMiAhuiufmedifmf^t^Mfti^ikttiM^ditttklteircMU* 
iniiMmtnfim crefcentitreperiri^StdntehtcdicimMs, 
• ^ V M Si 



Si graue,qu6d petpendicularit^ <krcendic» cewtnim .tame 
cdndngere poflfet , reUqua grauifteodan fimul ttmpocc fiogih 
la in Aiis planis conaiieicerent • 



\ ■ 




Sitplanumpdpendiculare d^/mff^ovfr 
ninoeritccnttitthcerrflet Si(ematqri»ceiitr& 
, ^>&fiatcirc&Iu$ ^«'i^.qwtranTeatper centrtt 
terrje ^,&perpun^m 4. d<^«auiadigre 
diuntur • Pofito deinde quoiibet i»buio 4 ^ > du 
catur i f . Oflendit Gaiileus grauia eodem t£- 
pore ad ^ , & ( • peruenire • Igitur eodem tem« 
pore conquiefcunt omnia ; quia cum fit te n raec em i^ u m i^ 8c li* 
liea tc perpendicttiaris ad planum dCteat r« pundum infi« 
mum plani ^r; ergo fi aUquod grauevlteritts procedere^ alcen 
deret .Quod eft impoflibilei Ergo&c»SuppOfiimusquod gra^ 
ueiliudpertinensaacentnimterraeftatimqutefcat» quod am- 
biguumeft&c# 

$1 cuxulum in eodem punSo duo circuli interius > & escteri* 
tts ccmtingan^& per conta^bm dua? re&x linea? agantur, erunt 
interceptfinter duasperipheriasiaeademrationecttmincer«- 
ceptis in reliquo circub o 

Sitvtponitur;&contadus fit 4. Dico eflTe 
vt^r.ad df/m 4/ad 4^.DucatjarAi.tan 
gens , qu« tres circulos contingat in 4 & iua 

Erit anguttts quiad/. (in alteraofegmen 
to ) aequalis angulo 1 4g . hoc eft ipfi %dd 
hoeeftvtriqueipforumadAy&r.Sunter « 

go #c> ^^>^paralldiat#Quarevt ct ad ri/ intercepeac iSh 
terduasperlpberias>ita ^4 ad tff^i hoccft 4/. ad ir^ Jnter^ 
c^ptffioceliqaockcalQtQg^ ..,% 




,% 



fXO 



r 



'^f 1. ! 



Lsier Minui » 



••^ 



/-^ 



*'7 




producatur 



fMOPOSJTIO.XV, 

SI Planadiuerfiinod^indinataadvoumpundttmconcurr 
rati^& grauia dimittamur eodem fimul temfMV ex aliqua 
circuli peri{moia*cuius in&num pundum fitoMQCurfu^ plano- 
ruin 4 ipfa grauia feniper in aliquo circulo (imui^iirpofiia comr 
meabunt. j 

Sinrpiona ^^^ perpendiculum, & ri • vtcun 
qae inclinatum > quae concurrant in ^,^ per c5 
curfum ^ tranfeat quadibet peripheria bcd. ica 
vt ^ . ftt infimum pun(5li4m ipfius * Dico grauia 
cx ^,& r.eodemtemporedemilTa femperin 
aliqua circuliperipheria cbmmeareiqux uran- 
featper^* 

Si cnim ponamus graue 4 delcendifle vfq; in _ ^ 

id. aequalis ipfi4/»&per<.&per^ agantur duo circuli qui 
priorcmciculumcontingantinpunAo L EritperLem.prasce« 
dens vt ae, ad cf.itz t^d.M ^^.ergo f/^8c ^^.funtaKiuaies. 
Sed ^i/,^^.exquietein i. eodcm tempore peragunnr, er« 
goetiam^r, r/.eodemtemporeperagentur^^funtenimaKltia-* 
les lon^itudine^&inclinationeipfis id, 4^.)C^egrauia4^ 
r, & inhnita alia demiila fimul ex peripheria^ c ^tfemper in ali« 
quaperiphcriafimuldifpofitareperientur. Quod&c. 

DAds quoicunqs fpatijs • deinceps in dire^lum continuatisii 
viiicdq»fu«ktionistempU0adfoti>ere«. v^ 
Sint fpatia quotcunq; dein^ 
ceps , (iue aequalia fiue ina^ua- 
lia^^s Jiriri/.vtapparetinpri 

ma figura & circa diametrutn^ 
4 d fiat parabola 4g. ducan- 
turqs orditfatinl ktycf^dg..^ 
£ntparalle)a?<iiametFo tk.fi^ 
^i^ 9 >6 i, i ^ tempoca eife 




(patiorum rc^dlue ditbc^td Hoc eium patet. Nam b «, 
fe ^/96 . tenipus-eft jpfias dh\St f^. t^ei </^. tanpus eft ipiius 
«r . quare Ji^i tempus eft fpari j i ^ . &c. dt &: de t«lic|Di$. - « 
Ptteft «ris^finefdrAoUiiitm.faftiktc imtd^ .Hf.fiftfe^ii 
.ddfigur*^ •'/Mmmt a e dqH^lii iffisk b:,ffim9^Mffwi/mi^ 
ilmiedt/0ii fyfw^endicMUrii Md a ^fi£idn»dcfmictrimU^ircit 
dianutru e b, e c, e d. ^^*-. quifecent rlitdm a Otififcn^is ffi. 
o. />«•» z(tii,io,ejfereffe{fiui tempora qumfita fpaticrum 
3i\) , b c , cd . (^ir< ItjimfifcHMmtiimfmffr^h ^^ffe^ihoc 
it^ i.irerittemfMsfer 3.c*.mediafr^9rtionMiis,^jOi4j^iai.cr!- 
W iif^ftitdiferetitif b c ,0rie (i '; ctdimmodte^tttdicmr um- 
fusfcr cdyejpio^^afefat4tJij^^ ., . ; . ; 

■•■'..'■.. ' .5 
Pmftfnumftt tmbisifmiehimfiAiime^arMUfibrum (qufdff^ 
jme yj^ingenioseieferit Galiltus) ^.iuerfo modo •cinjidtraik^ 
^et^nfm vt noi/isflus lucis ^irnt , ^ in^tm.infingidii.fa- 
raifiUftmSHs detcrminandos , ■6'-*iarint..ine»Hi conciffenif*. . 
'fUcnit -has^o^ofmont^ftA titulo J>:rmttu acceleraio 'fonero , 
AicetfafiMtjdiqtdddtfroioa^ioney ^uiafiinli^tca^fariAoU de 
^ener.t^earnmyqnadnitium iiAem-exiffertke^ ab iffo mtttu ni- 
^iiraliter acceitrato dttiwmt ctmcipnntn^tfmevila inMmmm 
iinumimftlitntiumoft^ . . 1 

^imatiUdHquodexAf^odom^^^^oidcm K > '^ 
aen^efoitagatJuoffatiai^hi.bc»£>i^iffmni p 

^emninotranftrefn^funiinmcifuantMfiiinng ^ ' 

rdefoKifftritjracedtneilofiimie.rranfeatdnimfi „ y, U 
fofjibiUjtnfer d . «r^tf ^«£tf ^/ij^J^ 4^ a. m^ . J»C J>. 

^n^ereeonfecitffatiazhyi^hdyfmd^cdn* . 
^dfilffefitnm . ^■enim umfotu cgnfieft zb fimMfiimh^ 
'^fedjfffamhc. ^uane conJUt&c « . 

P :« O P O S IT J ^ril^^ 

PRopofita quaiibetparaboiacuius v^xtoi&Mftatctm^ 
.aum^iguod fublancrepciiK5,«^ %i«le4J8dat ▼% 



iaiif,idtclBpuo£to f .iCtiitiifflpetttte^coBce^, horrizoofjb-. 
Iiiar$oniieioacfu'.»ijwiin profoiUiMn parobol^dieiacjbjUT' .'-, 



i . t 




pefitsparabol^* SiHDatur ian in fftrabolaquao , ^ / 
mnuisprodudaaliaiiodpiin&^ «. Dicograr . / 
Mpoftcafmn perii/» hbiiu^ontaliceiiconueiyriml 
ifi ^ ; cttm ioipetu iash conc^ptOrper tpltim 4 punSumtcaft 
fett^ Dcbetautempeft ^dzontalcm conutinofi^ 
&o t fai^am,grauitasfuamdcrccniIisofe(aU(Kicminch9^ 
Ducanturordinatim ch^ ef. Ponamufq;tempuSjCafuspcr^iu, 
^{k: cb. Ereograuchorizoataluc^conucrrumin ^,decurrcft 
mooi an^uat^i&tepore^fiss duplum tpfiMs c:)S^ rpianiim«hoc ^ft 
lenipqne ri'.ipiam 4?4»V€itcmpoi^/( ^c^dtai veJociGafcJ!^ 
ipfam/V • Graue igitur impem per ^ r , (ii^ pcr. ^^^acijpi^t^it 
conficit hocizontaIem/> • tc mpore /> .. »; fafMr, 

Scd codemtemporejfii^decurritcnamperpendicnihrefB if 
rquandograuitasincipkc^eraviin^ v:tincafunoIlr<>>erg^^ tet Umi 
dc ^cpocc coniurit tf^ &/r^Quaiic ^giiw.onininacr^Bfitij pj?r f ^ /riecri.. 

1 fanfirergo^gn^ poftiOiiw^ ^i^ |>tt ftigula propjof^ ;pfl!r, 
rabolapun*<aa.. ' - . A <.^.-:h. - ::. : ;./. .. \^».^.;i!-; 

''•'• ■••■ "^ •••■• '■"' •■■'•' • ■ • .• ■:'.•.. -J^k.. '...•■', 

ocdiaaftimdittritui^pn^KMadkultrfm^verof^ ^ ^ 

•ilnriigiHRi pii2fccdctiti4Fiib!po{itioJni»..^;^ ^ «^ : i , <o-r ■! ,. - .,^ 

vtraq^quana parti lateris.recti.Quiaimpctus funt.vt||gpr(|^it.j>-«Mi^ 
impetus^ad^s ex Ain btdwt ex Ain c,vt igfu «A^^i Pnogontiu- ^- 



iio De $mm^£rdm$m defi 

dumin parabola» impetus horizomalis in eo; erie vtcJk. Q^oeh 
doquidem impetas liorizontalis eft indelebiiiter aet]uabilis • 
Examineturiaiii piau^Hri quodlibet ^ • ln^ietus per pciidicula^ 
risquieftin r ,eft tdem aciifipetusnaturaliter cadentis per^yf 
ex qufete in ^.Vtero; enim dekenfus venitab altimdkie ^, vbi 
habent initium accelerationis • Impetus autem cadentis ex ^,m 
yCeft/r.ergoimpetus perpendicutjuitfiiii pun^ e.parab^iat 
ak/e^ Quare in eodem pun&o parabolas funt dpo inqieus^i-^ 
ter vt r ^«qufeft exfoco, aitervt/r^qpeex punda examina-; 
t6applicatur# Qupd&c. 

w 

Schoiium* 
Hinc poflec oft endi demonftratione dit^a prbpctecas tan-f 
gentis parabol^ , (iue Theorema mauis ^ fiue Prpblema pQfitoi 
priMshocprincipio/ ' - : i.' 







\ 



Si mobile aiiquod ^ inprimafigura 
ex anguloparallelbgrami alicuii^ , vel '■ 
ex quolibet pun<5h> diaifietti fctmir^ai^ - 
quaoiljtefduplici (imul l^ne, ti^pt^ { >i 
progrefltua fecundum lineam 4 r» & Ia« 
teraU fecundum a^ vtcunq» inclinata. 
fitqiproportroduarum veldcitatumeademac proportio late* 
nmdc ad ^ihomologe. Dico mobile iturum eiTefecundii 
diametram ad hoceft perip&niid^ttBMi^ \ ^ 

Si enim poffibile eft fetaoir mobUe^xira diamc^^ 
quod pundum/ » ducatiirq; e^ parattela^Ki^tf^ • Ergo quapi ^ 
propordonemhabent fpatia perada i^mofaiiit eam Iiabebimt 
&impetns: iiempe vtfpattumprogreifittumperaiftum^^adla 
teraleperadum ^r^itaimpetuspro^elfiuusadimpetumlace^ 
ralem>ideoqm ^g ad :ge ita ^^«d di ob fuppo(idoiMrm>£» 
tieirrad ri/^fiue ^^^;; ad;»';e(Ieiirci^a^le<2i&^^ 
lumdepars; 



t r' . •• ."v 



V^* ''•*''/ ' "l .** 



Efio 



L^er Wmiis: ^iit 

Bfloiamm^fecttnda %ura quodltbetpundum d incurua 
parabolica ^^r^&appiicata 4^,fa^fq>aBquaIibus Jc^ ce^ 
ducatur ^r^quamdicotangentemeflTe . £fto focus/1 & ap< 
plicata ex fbco vt&zfb . Erun { iam in d duo impetus alter pro* 
greffiuus deorfum fecundum diredionem iinea? cdy dlter kce- 
ralis fecundum i/4,eftq;progrefliuiimpetus ad lateralem ra- 
tiovt ^^ ad //iperprafcedentemPropofit. fiue vt cd ^Add 
( cum fquale fit redangulum {whed^ femiflfe lateris xt&ifl^ 
quadrato dd) £rgo njobiie dum eft in pun&o d feretur iiecun- 
dum diametralem ^r ^ ; fed fertur etiam iuxta parabolicam ii- 
iieamquapci^curren5defcribir,ergo reAa dc^ &paraboiica no 
f e intcrfecant in pftttdo d » fed tangunt .• &c. ' Harc demonftra- 
uo pecuHaris eft-^oparabola; fed & vniuerfalem habemus 
ptbquairbetfediioneConica> confideratis a^qualibus vdoci- 
tahbus vniuspundki^qttod^ualxtermouetur in vtraqdinea qu^ - 
exfodsprocedit. . , 

Eadem ratione demonftratur Propofitio i S'. de lineis fpira* 
libus Archifnedis vnicai breuiq; demonftratione » non foium 
qiiando tangens confideratur ad extremum primas reuolutio* 
nispun(5ium]fed vbicQnq;punAum fit incuma fpiraJi femper 
oftenditur periphaBriaiquasper pundu contadus ducitur f qiui'- 
iis cuidam redae linea^&cQufle Propofitiuncula cumoiim ioter 
atnicos a me vulgata fuifTet , Clar. Vimm Galileum meruit ha* 
bere laudatoretafi • vt extaat ipfhis epiftolas apud nie » 

Immo & hac ratione oflenduntur etiam vnico Theoremate 
ta[ngentesquarumdamcuruanim>interquas»omnium iinearu 
Cycloidaliuin^ytb^^euiterattingemusadfinemlibri deQua* 
dnituraParaboIae^omittentes demonftrationem tam tai^en* 
ciumiquam etiam fblidonim^&ceiitrorumgrauttatisipfius Cv^ 
cloidis ad euitimdam molem. ^atisfit interea ledorem hic ad- 
monuiirequ6d fi Cydoidis fpatium circa bafim conuertatur» 
erit folidum ad cylindrum ci^cumfcriptum vt 5«ad %. fi uer6 cir 
caiaiigentem bafi paratlclam ut 7. ad & Centrum Cydoidis 
axem fecat ita ut partes fint ut ^.ad % . Demonftratur etiam ra-- 
do fblidi circa axem ad cylindrum circumfcriptumiitem in qua 

Q^ linca 




linca axi parallela fit centrum femicycloidis . Ciar. Vir ikito- 

nius Nardius oftcndit qu6d fi Cydois circatangentem ^ipa- 

-■ ralleiam conuertatur folidum ad fuum cy iindrum eritfubfe£]ui 

tertiumiquaromaia iortafle aliquando edennir,interea ad opus 

reuertamur. 

Si inHiametro paraboiae aequales fint 4^ ex vcrticc tScdS. 
fion«x vertice» Eritquadratum ^ r . aequaie quadratis df.cg. 

Sitz\i. UtMSTe&um , & comfUdHtm: tt^dmguU c b , d h J 
^U a c • db . pnMHtur fqMles^ trit .j^ 

nifM^MUmhli. dquale dHobusreifdtfgm^ q^ |y ^ 

Us dh>ch,/riy {qnodidem eB)qH4dr0tMm 

tb . dHoins qHddrMtis dt.c^.ffHMleerit. 

Sijiod&c. 

PROFOSirtO XJX, 

tMpetus in punftis parabol» funt intcr fc vt line» ordinatiiii 
applicatae non ad ipfunetpunda » fed tarito iongius a verti- 
. cequamaeftquartaparslacerisre^* « 

Sit parabola cuius vertex d. (ocus/. Sumpto 
que in ea quoliberpun^o ^ » Dico impetum i n^ 
r ciTe vt de^ qua? applicata fit tanto longius i ver 

- tice^qu^mipfa^^^quantaeft ^/loempequar J^ 

' tftpat^latcrisre^» 
* Impetus cnimquiiimidfuiftia c^ fuoit tbyhf^ ergo momcn 

lU^Iimt ^^^'"P^'^^^P"'^^™P^^^^^ ipfis^ua 

m 4imi: k . Sed & rcGUde . ^qitatur potenda ipfis e ^» hf.^tx le(n- 

ma pra^ccdens > ergo momentum de .. eft momentwn fiue im* 

peti»c6pofitasexdttobjasiiUsquiiimtio pun^r. Qu^^&c. 




« " 



'• . .V 



fMO' 



/ 
I 



r^t 'kid i^& s-rr^ f^^o "^x X. 

I 

IMpetusinqn<)fibetpaniboI«]i&^ideiaeftaciinpje^ ^;^ 
uisiuitnraUt^cadeniisexfHbliilauaiefiiBul» 9c altinidiQf^ ■»»«#«• 
eiufdemparibbl^ . 



V .• 




Sitparabola cuiitt*iJdtudo r^^firf^limitas 4 
t . Dico itnpetam in pun<fto h eundcm dfe ac 
naturalitir cadentifS ex 4. in r . 

Sumatur r/. aeqiialis quartf parti laterisr re^ » 
hpc eft ipiS 4^Eritf nipetus in pundo ^ : ut Wtit^ 
r/CperPraecedtotem. Atimpetusetiamcadea- 
dshatundit^per ^/yfiueper i^r» efteadi^tiiiinea efi ttf^o 
idenvimpecus eft in pnndo /ft parabola; > acmpun^o i* « grauis 
debpfi ex fublimitate (imul & aititudine db^t^c. Q^od erat. 

p i P S I T I O X X /.' 

t 

TEniporaladonuih perdatai(}10tfa$ horizontale9pb1Fjca<^ 
fus h perpehdiculo, jfcqualia enint » quando akttudifiet 
perpendicttbruni dnpiicatam raitonem hamKrintiUii» ,-iqaani 
horizontales linea; hab em . 



• . I 



iUiusqoam^» 3_A \ 
r4^,eodem . \7_\ 



. Sint horizontales Une« dat« *t,cd,9i9X 
dudines perpendiculares fint 
e 4 . ad/r. m duplicata ratione 
habetad r ^. Dico poft cafus td ^^^ 

temporeperagi 4^,de^i/« j . :]9( .' ^ 

Hpcaptonpatet.quiacumGnt ^ihicd.va ' > 
rubduplarationeipfarum esyfc, ensitetiiun>t temp(»a ca- 
iuum,5cideo vtvelocitates, fiuevtimpetasquifuntin4^& r. 
ProptereacumiSntvelocitates inr, & #»vtipfal|KUia cd^di 
homolog^eodem tefnpore^fafpadaperagentur. QKKi^» 

4 

C^ a ^ Pita. 



\»4 De m6i^^i:4f0Uf9$ , dffcendent. 

JLj ^crcun^uiuucanxtbili.sjqiij^ e^ coii^enatw Impe* 
mpriusaquifitopercafumdiametri ecquiete.^^ai^^ . 

Sit 4 ^ . ^iameter parabola?, & ^ ^9 ^ e • ordi- 
natimduaaf. Dk6m^,^Qf^^\m^ 
^i/»&poftcaiumiir.ipfaQir/^;.a?qualibusjteraL,, - 
poribuspertranfire* . -. . 

Sunt enim altimdines peitpendicular^ dl^d 
, r.induplicatarationefpat^orumhorizontaliuoi^ ,\ ^ 
_bdyce.o\> parabolam« . Qffiire per prasceden**' . .^ 
tem eo^^cempore perageiitur ipfa . fpatia,:(ior^pp£alia ^ boc 
efiipfa?ordinatimapplicatacipoftcaiij$ 4^i|4f «;|Quod &c* 




? » 



* \ 



FROPOSiriO XXllJ, ' 

•^- ' '■• ^' \ t \ .', . \ O, ;V <-, 

t 

4 

frt^ £m^ora lationiffii qt» £unt^ veniceperxliametrii^ 
V :X' tiones fimul & fuas prdipatimdu^as, funt ytipfa? ocdi- 
naiipi<duda^iaddita.tamfnii0g^ m^diet^te late^risrediJ' 

Sit parabola cuius vertex d , focus h , ordi 

natimex foco^&^&icii4i q|ia? a^qualis 

enttlatet-i redo • Demobfirata enim fuit ipfa 

c h^jMit h iMupIftipfius hd^&i ideo fubdup* 

la la^eris reAi . DuciBir per i/ parallela dia- 

metro 4^/« Dico ten^us laitionis per A^^ 
f« gf>^]^^ * €i%tXk\U.ih ficde fingulis . 
.iTcnipuscafuspcr^iA eft^f^c. fedcwmdupla fit^^. ipfias 
4^.temptts per ^r»idemerit, ac ^i db^ nempe hc. Ideo 
«r « tempus eft omntum bf. e i . &c. ( cum eodem tempore om 
nci peraganmr per Prae^dentem) .Tempof a jUitem caluun^ pcar 
db^dfftunt ipfae bf e $ . Propterea tcmpus per d bf. erit fb^ 







\ t 



• * f 



B 



&i i^i ; &stul^vtlj^ < jrcf^jH«tein per 4 rvi • erjt J^c;fia»|il cum 

Aliter idem oflendemus» 

»/} ba / i)^^ itnmuttmfw. kH^wi^ .i^ ; . ^;/-'^^ ' 

a b c ^ b c .s^umJ^tmijg^Jdt^isreiii jdr jh , u/T p» 
Pcdeji^gu{ifj^ Temfus fuimfer a b a ^ */./,' 
bc. c^^^ \>Q.dufUmdUitudwisfo3^^ ■ - ^^: . . / • 
ca/um ab ^ttmfus erit ekdem bc» C^r/» ^ ^ ^ , ,. . , 
itdq^^^mptsfef. feb-ii^/«^ i> • ^fV^ftfs ftt c b. tdift\ minus erit^ 
fUdMj^/fdiium:mi0^4*ej^ s fum id^m imfffHs retinedtur * Sfi^ 
mdtur ergt.if/drHtfi ^^^i^^^AmM^ &erit f 

b, temfus ifjius c b ^o^cdfum a b . 

CdterumJtinedmL ihcfjk/efuiffefft Idtmsreifi^cnontmfd^^ 
€imus . Heifdngulumfub Idtere reCto , c^ a b . fquale e/i felid^ 
gtA^ CL b £^^virum0A$f9i 4^dtur:qud4rdto b c. ) ^^^ifr^ ^r<:£f- 
_ f9ifH$dlfdkilfuntMt^ds\h^fif^ ddhsi.jMh^ftfU^ 
ifdddtu^ re&umidd\ it^d ilTiHs^/uhduild erir. Ej^itd^i ^^ 
pusfer a b c . ///2r B c . cum b i./emifeldteris reSi i ^pd <- 
Pdt&c. * ^ 

,'. ^SrtfdtdMM abc» ^itc^frtic^eincUnetur -- -<^^ 

a c . e^ ordindtim dufidntmt f 4 jtxfunRa tc«. 
«^ b e v/r umpffcden a.<i^inj. . . ^ , j, 

' 2)/r^ b e . m^ijdmfrifartiifldUm effe tn^ ^ ^r. ^ ^ ^ 
/^ cd , c^ f e * ,£>? enimd a i^ a^i Wc d* 




^ fc Jf dufjifdtd r4Hffti:i£fi^^^ 



^dremedidfrofortiondlis e^ht Jnter iffdi 
iTdt (jrc. 



p M q:p a.s IX i o,..xxjr^ ,\.,, 



T- 



.> !k 



■V . » ' 



V\ 



* hbrizontem incliQatae/unt vtlim-^gi^^perf fi^^iL^ 

mor- 



. • 44 < » 



^ \ 



^ brilifikltiftfdtictntar iti pitalb^ki» Cttius ' ^iaineter fir H<»i< 
zon,venexauteinpunduinindinatkMQll. ''■ 

> •• ' • • • ■ 

Sit Knea ad horizontem inclinata ^^ V « 
fithorizon iir.^&circa diametrum;4#r^<!S^ 
^fcribatur parabola quse fecetii ^ « in qQ^iibet 
pundio ^, &ducatur^^.adhbi^i2toii«eftu» 
perpendicularis ; ^manturcf) {>unda qtiadi^ 
bet ^ • & ^ ; ac ducantur ordinatim i^ ^ > ^ r 
/i. JDico tempora cafuum per ^ h , & per r/? efle iA^^d/. 
ehimponamusteihpusper ^a dk i/,^ttcmpiisper gA.mc 
diz proportion^Iis iJ^;Scper tf. media pit^onionaits df. Va 
demonftratum eft in lenimate prsecedend . Quare &c^ ^ 




ATH 



• \ 



p R o p o s I T I xxr. 



SI in parabola lih^a ^ ^ ex vertice inclinemr » 
ducana: de^ quainclinatamfecet in/l ^fitipfit-^/^rem^ 
jpuspcr ^/.&reliquay2/.tempu$perreliquam fh* Quando 
motus veniantexquiecefemperin 4 • 

Tempora cnJm per df^h . funt in fubdupla ratione /patio- 
]iun//',<(^.,(iueb'nearutn je^Ac, SuntideotenqK>Yavt'r^, 
edyi^ quia ift$ funt in illa fubdupla ratiotiie j vel 
vc «d, */; (funtenimconanuf r^,- edf ef) 
Qiiare cumtempus per df. flt ef. 6c per /t6, fit 
«d^ etitfd» nempe reliquiim tempus,tefflpus per 
/^. i:e%wttni^atiumpoftquieteiAiD4;Quod 




(. \»<v 



Lemma> 
Imetiitetmrtx^oertieef^or^dfre&d ib , (^pj^MMt tdm^ 
^dtinvertieere^d ac . Dtu^Mnr qitdms dtid ^l^qMdtccttrrdt 
fdTdhU iit i i &iiteUndt4 imc*. Z>ie9 d c . medidt» Prtpmi»* 
^ditmejeimterchtdi. 




Efiemmhc^Ad^i Ungitud^t *vt cz^ ai 
ad» vflbcdd At.f$temtii.^tM4te ineemti^ 
mMMfrepertiemefftnt b c « d e .9 i &€.(!rAtMe 
diseSt^ ^gedefertekM efiendere. 



PROPOSITIp XX-VI. 

SI Horizontalislineaparabolam conangatstempora ca« 
fuam ex pundis parabol^ vfq> ad horizonte \ (iue ex pun- 
&x% horizonds» vtin fecunda figura » vfq; ad parabolam \ erunt 
ut lineas parallelf inter horizoiteem> & quomlibct alia^m ex con 
ta£tu inclinatam intercept; • . 

Sit parabola bde. cuius uertex c ^^ ^ ^ ^ c 

&eamin^.tangathorizoh ed. Ssc 
ex contadu ^ • ihclinetur utcunq; c a 
ducaturq;expuni%> d. dd. horizon 
tipcrpendicuiaris • pufia iam qua- 
libetperpendicillari ef. Dico tem 
puscalusper eh. t&tfe. &c« & (ic de fingulis » Po^amus 
tempus per dd. efle dd:^ erit tempus per ^ ^ • ipfa fe . media 
pro|:ortionalJsutidemdnfiratumeft» Quare&c 

p K o' p o s 1 T i o xxrii, 

T^mpora lationum per chordas ex uertice parabolf jndi- 
nataS) compofttam rationem habcnt » ex ratione lon^i^ 
tudioum chordarttmj[^ e;Csratiane ordinatim applicatarum»c&» 
trari^.tamejii^imptanim « 




Sint chords ex uertice dhy ^rr.&ordina-- 
rim^lucanair h d^c e. Dicix tempus per ^ii^. 
ftdtcmpvisper dc. habere rationem compo- 
fitamexratione dt ad dc^ ^exratione c 
t ad hd^ Sienimconcipiamuslationesil-^ 




las 



""t^t De motu ^amimdefceHdent. 

las accelleratas ixquabilesi iieri«^& gciHiia per {p^atia ^m^cm 

rcciirrere cum grdfdu fub^duplo.impetiiis quem kjtbebincin '^;& 

i.Gai.dcc^ erunt tempora recurfuum eitiem aC tempbra caiimm . Ti^m- 

motuac, poraaulem lationum aeqiiubiliuth xompofiiafti rationem ha- 

ULde mo bentexrationeTongitudinumfpatiorum 4^, ad ii^; &ex ra* 

tu^qua. tione velocitacum contrariefumptAmta*^ ^aAi^. (imi enim 

velocitates in ^ > & r eaedem ac in ^, & r > & velocitates in d^ 

- & € funtvtt^mporai^.4r^.)Ergo^riamtempofacaruuiiitta- 

turaliteracceleratorumper MbyMC* compofitamraoonemJia- 

bebuntex i^fdem rationibus ^^.ad^c.&^^.ad b4* Qupd 

erat&c. • 



^i -. 






pROPosirio xxrxii. 

TEmporaiationumperciiordasesruerQce paraboi^> fdiQt 
vt tincoe ^quse ordinatim applicantur non ex termiois 
chordarum» fed ck pun(^s diamiecriin quae cadtmtHaese redos 
angulos continentes cum ipfis cbordis^ 




Sitparabolae diameter Ag . & chorda? ex ver- • 
tice (int db.dd. iiantq: anguli abf^ a dg. redi, 
& ordioatim ad punda fg. appUcentur fh^gi^ ^ ^. 
Dico tempora lationum ^txab^ad. eCfe ipfas or h; ^ 
dlnatimapplicatai'/i&>^i, ^ ^. ^ l \ 

Tempus enim per ^r ^ . a^quale eft tempori per ^J 
af. exiftente angdo a bf. redo • Item cempus • 
per dd. ob eandemcaufamaequaturtemporiper 4^.Tem- 
poraautemper^/.^j^.funtjpfe hf.ig. ErgotetnporalatiD- 
numperdiordas^^.^^.(unti&/,&^i. Quodcrat&a 

Proponetmr ttidm hpc m$do . 

• # 

# 
Tetpporalatiofiumper chordas ex verticeparaboie funtvt 
Hneaequf applicantur non ex terminis chordarum , ied tanco 
lo^ius a vertice quanta eft laterts re^ k>ngitudo« 

0/lep' 



tUer ^rhmii^ 

cpttfmn emm tfi imfirfetdenitjtepiuf» a b» 
(fitBtdmiilk^hc reif9)ip tiMeMm cA.Ok», 
temnc Ume4m d c tdmte lemgims i^ertiee 4ffUeA-^ 
t^tmeffeytpnmmiffd b e, qtuuU4 eli Uueris reiH 
UtegifmHe . Hee efi ippm e c iMms re&mm effe . 

BecMtet . Eft emim re&dmgmlmm c c a. <(f M* 
le qmddrat» ^b . «^ mtffttmmreSmmsdh^freften 
redmmeft, ^gfderdft^e, 

Corolbriuin. 

jqium$Uhttftn€Umdidmtxv€fHciifutd ab, 4r 
fiftionimdxiffiHTiffindintim ac;. cui tmmm 
ddditmnfkifitindifi&nmUtms nifmm a€» itd 
n^t Utioms fidiit fir z\s ix qmiiti imz^ &fir e 
c ixqmiitiim e. 











PROPOSITIO XXIX', 

TEtnpora lationiun per lineas ^uae ex £bco parabolc indi 
nanmr* funt vt Uneae ordinatim applicatae non ad punda 
tn<juaecadunt inclinataiuin perpendiculares, fed tant6 ftVO; 
rius verfus verdcem , (pianta eft quarta pars lateris re^ . 

$itparabolacuiusvertex4* focus i, Sccx 
fbcb indinetur ic* fiatqjangulus 6ed, redus. 
&dpundo i/.fumatur verfusverticeniparabo» 
laelmea'</«.aiqualisquanaepartilateri$ re^. 
Dicotempusper^r. eflelineam e/, Tempns 
cnimper ^^.apquaturtemporiper^</x>bangtt* 
lum^r^.redum,hoceftper4r.^funtenimaBqiiaIe$ id,de) 
fedtempusper4e,eftipla«/.cfgo(ciiipusperii/, yel i#» 
€mcad<em e/, (^odeatCec, 







a 



PJtO* 



qip^ De mkiiwifdkimdMfcendenf. 



r^ M B) ^-^Si^ t> f ■<>■■; -Jrjr*» 



D 



/» IV 



Ato plaao kicliRatb perpendicidum €ti^ffts<f»odto6eM 
temporcac ipfuth plaflum inclinatum<H5nficianir . 






A. 





. . Sit ioclinamm pt anum 4 ^ . cuius erenatio -^^ ^ • .j^ 

fi .t vt 4c. ad >4^. ita 4^,adaliam; qa^ Rttil^ 
Dico planum ^ ^ . ex quiete in ^ , & perpendicu- y 

km ^/r « ex quiete in ^.eodem temp6re confici . ^ z^ 

Tempus enim per ^^.ad tempus per 4Cy eft 
vt 4^ ad 4r/.temiiusetiftint>er i/r.^^dtempas ^^^ 
per 4c . eft vt 4i i^mediapfoportion^fiys ad m^; quar& tempo« 
raper^^^&i^^aBqualiaerunt. Quod^rat&c; 

P Jt P S I T I O XXXT. 

AD datum perpendiculum planum infle(5lere data? longi- 
tudinis > ib n perpfcndiciUiAn i^film ,> &finflexum planu 
codem tempore abfoluantur . 

Detet dHiemJongitftdo d4ti ft4ni minw tj^t 
f^oferfendicHlf^ T jAi 

Sit datum perpendiculum ii ^ , & data plani t^\ 
longimdo fit r ^ minofperpebdiculo'*^ Fiat vt ^ £ g 
h. ad r . ita r ad aliam quae fit ^^ . &: e Jkr pun^o 
d aptetur de . oKjualis ipfi r. Dico tempora lationum per d€\ 
Scptr46 effe acqualia .. Huius demonftratia congruit cum pr^ ^ 

cedenti>quando^demiacontinuapropottioneiuntii^> de^ 
di. Quod&c. , 



• ' \ 



' ' ' . * • . » • 



M 1 ... 



> M > 



f ji p o s i T i o: XXXII. 



Xjl 



I > 



I> datumpetpeDdicuIum 4^, pranuminffedere ita vt 
cum per^ndiculo quemlibetdatian angulum acutum 



-" :v /* con- 



.'j'. 




contineat^ puta.ci)uttkm ipfi ^«m^^ eoden tenipOKe acipiiim^ 

perpendiculumaMdluacuC'4 

* 

f idt circa diamotnim iij^rdroilHsqui fecet ^^ 
in ^/.peniifsaqjpeipetidiculjvi^^.campk^^ 
paraUelografnmum. ^ ^ r/« 

Manifeftum eft planum/V • qusfito noftro fa* 
tisfacere. Cumenim/V.^qualis fitipfi dd. & 
a?qualiter inclinata ob parailelogrammum , eo- 
demtemporeabfoluentur/r^^^vel^^. Quoderat&c. ^ 

, ^RaPOSJTIO XXX 111, '' -» - - 

AD datum perpendicuium dc^ in figura Propofitionis 
X X X. & ex dato in eo pundo 4 . planum infle^iercf 
quod codemtemporeaoipfum perpendiculum conficiatur ex 
quiete» 

Reperiatur inter dcyCd media proportidnalis dby Sc ha« 
bebimus longitudinem plani alicuius . Applicetur b^ iongi- 
tudo ex 4 f (itq; illa iam applicata 4i: Manif eilum eft ex pra;« 
cedentibus Propofitionibus ipfam dk. &perpendiculum dc 
eodem tc mpore abfolui • cum fint ia continua propoitione dc 
4^4r. Qyoderat&c* 

PROPOSITIO xixir. 

SI ad perpendiculum aliquod d b , planum c d inflexum iie 
ad angulu femiredum»Erittempusper r ^.a?quale tempo 
ri perpendtQuli , quod ipiius ch. duplumfit.Pit>p(>netur etia 
hocmodo. Tempus perdiametrum quadrati ere^> aequale 
«iltemporiperduplumlateris ere^» 

Sitcd. fUMim vtfuffpfut» . veljit didmeter ^ndtkati.eit^ 
i0s Ut0s c b . €re0iim Jit .fothttmr^i a b . ai^d^iffiiu cb.Din 



• . .• 




]}a De mofsiiCrsmtm defiefulertt. 

ddc b poieniU eSvi a b ddeMmlemch . Ungi^ 
tudine , nemfe in rdtiene dt^U y erunt cdntinud 
frofertiQndes a b , c d , c b « ^ju»eferfr£cedet$ 
tes Prefefitiones eodem temfere nbfoUentttr fer^ 

fendicMlmm a b , (jrfUntm incUndtttm c d. 
^oderdtc^c. 

pROPosirio kxxr. 

AD 'datum perpendiculum ^^ ^planum , vel piana incl 
nare ad damm in horizome pundum # , ita vc indinai 
plana&perpendiculumipfumeodem tempore abfoluantur. 

Debet autem pundum c. Diftare a pim- 
^o ^ • no amplius quamiitfemiflis ipfius dk 



Fiatcirca db. circulus dedb^ &erigan]r 
€ e. qua? omnino incidet in circuium • ^aiias 
wcbienlainfoiubileeifet)incidatini/.& e. 
Du^q; ef. dg . paraileiis horizonti t c . 
Dico plana fc^gc. ad pundum c inciinata , 
codem tempore abfolui* 

. Cum enim eb.fc. fintdiametrifigura^re» 
dangui^ ereiftf » & ideo asquales ,& f qualiter inclinat^eodem 
tempore peragentur . Ergo tempus per ^ ^ , per r ^ , vel per fc. 
vnumatq>idaneft* . 

Eodem modo inferturtempusper^^ ^uale tiit tempori 
per^i, Quodcrac&a 

pRoposiTio xxxrr. 

IN datocirculocuiuscentnimeft ««.Diafnctruin appre iea 
vt teropusperaptatamdiainetruin ^quale fitcuilibet dato 
tcBipofl* y 

, .Debet autem daniro tempus miuus efletempore cafnspec 
- . > dia- 





IA& Trkmis : il$ 

diaroctram perpendicularem • 

Ponamus tempus per diametrum per« 
pendicularem cd.tw €d. & tempus 
datum fit ^^.Reperiaturipfarum sd^ed 
tertia proportionalis qu^ ^tfd .Sc circz/d 
fiat circulus/i6 d. in quo ex pun^ d. apte 
tur ^^•^quaiisipfir^Poftremoipfi td^ 
agaturperii.parallelai/. Dicodiame* 
trum #7.datotempore r^.abfoiui. 

Cum enim tempus *per # ^» fit r dt erit 
r dy (^quia media {vopordonalis eft ) tempus per /^» hoc eft 
peri d. ( per fextam Galilei de mom accelerato^ hoc eft 
peri/.qu^rauaiis&parallelaeftipfii&^. Tempusigiturper 
diametrum il.dk ^^.Quoderat&Ct 

PROPOSITIO XXXVII. 

SI fuerint ^^.^r.adhorizontemperpen- 
diculares» &fumatur.vbicunq} pundum 
€y fiue intra » fiue exn:aparaliela$ , fiatque ad e. 
angulus i/r^rc(flus»Dicoperinterceptas dd^ 
i c. femper efle tempora ladonum ex quiete 
equalia^Ducamrenimper d df. parallelaip» 
uic. eric angulus/Ii d. ^ualis angulo r; & ideo redus . Qua- f^f *»• 
retemporaper/W.^.lateratrianguiire^nguli» cuiusbafis >i^^ 
ere^eft, aequaliaeruntinterfe^ &ideoedamper bc. dd., 
( funt enim df.bc. latera 0{^fita paraUeiogrammi,qua? fem^ 
per eodem tempore peragimtur • ) Qiod erat &c. 

* 

PROPosiTio xxxriit, 

SI abeo<iemhonz(Midspiin^o4.adicleinplanumper{)ea 
diculare ke . dao pkna inclinemur ab utd ^ qualiter ab ia 
clinationefeniire^diftantia,temporalationumper ipfa plat 
oa indinata» fqualia eruntinter fe . 

Erigatur ex d peipendiculum s r. Fiatq; ^irculus circa tri^v 

Quia 




th^ jDe mota^fraiikm )iifcenient 

Quialinese ^b.Ad. per hypot; aEX}turlitar 4i-' 
ftantabillaqiMeanguktin;re^m ^iir.^ btfanam 
iccat, o^qualiterdiftabuntcoamabipfis <i^r A^& 
anguli ^/^c>^^r.^qualcs€Fi»it9fed.^4.rt &^^.. 
d. flmt altcrhi» ergo se^lcs erunt 4A c ^pcdi di : 
quare triaiigula rcdangula ^ct yddc^ «^uiaj^ii' 
laerunt,&vt ic.ad cdy itaeritr4.ad ri/;&; 
ideopervlrimam terri' EucL refta c<a^ circuhimcoiitinget. 
Scd catH horizontalis , ergo pundum m • eft pundum infimG 
circuli , & ideotempora larionumper «6^ v ^^ ^qualia «fu&t • 





1« ^ . . * \ 



Alit^r. 

flot idem ofiendcmHsfin^dt/^Mlo yCurinfd qudddm inMetfi§^ 
•ne . Sinteddempldna a b , a d fudmuis h c non fit petfendi^ 
culum yduthmodo inclinataptandfitciintati^uioscumJjotizot^ 
te ^.x.^^€umflano hcpefmutatimaqualcs^ikitctB cab A^ 
quaJemiffij3LdCy& c^dipfizbc. 

lam fofitum efi ttiangula a b c , a d c^ 
efiefimilia. Imaginemutiamconuettifigu^ 
ramitdvfhcjit hotizonydr ac faBafit 
fetpendiculum . Habebttntin iUofituflana 
a b,a d .ejfdem inciinationes quasanteinuer 
fionem hdbebdnt yfftmutmm tamen . nam 
a b minus decliue etit^ ^ a d ^ magis ; habe^ 
buntque flauA in tofitu eandem communem 
eleuationem . Etgofer 2 .Jfuius , wit in eofitu inuetJo Momen^ 
tuminciinationismaiotis '^d^admomentuminciinatUnis mi- 
notis a b , w a b .n^ia^ a d . J^efiiiudmusnunofigutam inptifti^ 
numy'&Jfabebimus ( fetmutaiispianisjeafdem inchnatio^ 
nes. J^icamusigitutitetum. Momentuminclinationismaio^ 
ris a b .admotnenru inciinationisminotis^,d .eBn/t ^h.adz 

d . Sluarexumfint^om€ntavtffatia,eodem4empor€Mtmtu. 
/jtrab.ad. ^^deratcjrc 

Poteratetiamproponific. Siabcoacmhorizomispunao 

■ - - — — • *t 




it duoplana adaliquociptaiium ^rinfledantiir^itavt 4^ ad 
dd Gtvttc adr^.erunttemporalationunQpervtruinqiincli* 
natum planum a^qualia^ 

PHOPOSITIO XXXIX. 

• f 

I 

• I 

St fuerit quodcunq; planujfn eleuatum 4 ^ , & quodcunqufr 
horizontale fpatium dc^ fe^um bifkriam in d. Dico ft 



ponatur tempus per 4 by t& m b . tempus per 4 ^r poft cafum S 4 
cfle femiflem ipfius 4 c^ nempe 4d. 

Ponatur enim 4 r» dupla ip(ius 4h. * 

lam fi fupponamus tempus per ^ ^ .. efle ^ ^^ ^^*^ 

4b. erittempusper^reademxrA.Sed E c? p H/ 
ii fpati j 4 r e(l tempus. 4 hy erit fpati j ^ r 
tempus 4d .(t^ enim vt fpatium 4 ^>ad 
Ipatiu 4 r, ita tempus 4 ^, ad 4 d^ Quare cum tempus per planu 
eleuatum 4 ^ • fit ipfa 4^ . erit po(t caium h 4 , tempos per ^r . 
dimfdia4r. Quodopoctcbat&a 

« » * * 

Hd^ Pr&pojhio re ifjk cMgrnit cumffofof.^^s.GdUlti de JI/#- 
tH4CC€ller4to , Hos ilUmdiuerfo modo frofofuimm^ ^onfmlem^ 
tes offortuniuti eorum qu4 hincfequuntttr^vt iufra afpdrehifk. 

. :-f Jt.Q f O S I T J O X Z.. " j 

l 

SI ex terftiinis 'm' &^v alicuius linea? horizontalis ^hio plana. 
inxqualia dd i<khi pundtum r. compofita fuertm^^^ 
tnaiuS >^itn9inus ^^&cH^e^nttfllongkudinis pianorum ^^ualii; 
fit fenaiffi hdrii2:0ntatis 4 h . Brit tempns lationis ^kc£\x ex ^ 
vfq,iS*>.*«^aW^^ c pcr if^^Wqfi» 

cund<flikterininum 4 horitontalis fpatij • 

V 

' IDtiitidatur'^^ .-bifatlam in /. erumergo ^^» kd.^ «q|ual«( 
i^^4. PoAanuisiemptttpert^r^e^^.^^^^^ 



•■ • • • 





f ] ^ ^e mm^ydmmlie/cmle 

cempus 9d ipfa r^y&perduas chtha^ 

tempuSiper.Praecedentcm, erit ckdi Dem* - 
pe (qiiale ipfi tempori c d . Quod erat &Ci 

Problema» 
/^/;yr mdnifejld eftfclum f foblemdm ; 

fUfjtm , inuemire cd.iia vifi t^ierminis a , d* b • nd^nnm 
funclnm dno ilUfUna comfonaninr, iemfns laiionisfer mafiis 
aqualefti itmfori Uiionis per minns t &lforizontem fimnl ^ . 
Hematnr expUno a c • fors c d aqnalis ipfi a 
e ftmiffi horizoniaUsfpai^. reliqnnf^ a d > w/ 
flanum quafiium . 

SiverofaCfadeiraCHone ex cdi nihilreli* 
quumfiiy velfierinullomodopo^ty froblema 
infoUbile erii . Demonfiraiiopatei tx Prdoe^ 
denii . 

Daio vero minoriplaM a d , (^fpatio hori^niaii ibito^d^ 
demfignra ^fiipfi ad .addainr d c.qns aqnaiisfit a e ^femifi 
htrizoniis , maiusflanum quafitnm erii a c . Debeni anitm v* 
traqipUna a d ; a c ^finml , «f^i^r^ tfifefpaiio a b , 4/^Wx infoU^ 
hile tjfeiffoblema ; Mini duo laitra iriangnli rtliquo dtbtni tjft 
maiord. 

^ando daiafuerim ipfa duo plana iufqttaUa ad,ac> & 
quaratur quantumfii fpoiiuin horiMniale^ tx cnius titirtmis 
funffis trigidaiaplanapojftniy dtadvnnmfunBtimctmfont^ 
itaviitmfuslationisfermaiusflanum^ aqualt fii itmftri U^ 
tionisftr minus (^ftr hmzanitmfimnl; Accifitiur dijfereruU 
fUnorum d c , quf duflicataffaiiinn htrixanioU quffiium a b 
txibtbii . Dtbtni anitm irts Unta a b » a^ , a d talts tjfe vt 
triang. fojjintcofintrti aliasfrobl. tJftiinfolnbiU.Htrnmom» 
ninrn demonfiratio cum ilUfracedeniis Propofiiiotds congtnin 
idto rtm indicaffe fatis duximus * Ubei hic obiter recenftrt 
quafdamfrofofiiiuncuUsy qnamqnam ex 3. Cotoic^nm dtptn^ 
dtatiffarnmdemonftMtioiaff^ebii tnimtxjfsndmrdmtiidmi 

circd 



€ircd byfirhleM quMfddm mgdsmediuidmfkiffi ddmemm/fe 
&4mtes . Si cui cenied mnftdcemtt Jigrefiene hde frdtetmiffk^ 
fjmcdbsceuitdref$terit^&dd frefpfitierfem 44- fe cenfcrre. 
Mdteridfrfcedentinm hdnc centimtdtiemem tmmis exfefimld^ 

PMOPOSITIO XLI. 

SIredaIinea^^.inquacitoraeqale$paRcs ^r>r^/,^/^,<^;di- 
ui(afuerit,&expundis r,r,excitentur duzhyperbola?» 
quae fe^ones oppoficas dicuntur> quarum fbci finc ^ , & j^ • Snm 
p.toinalter^earpmquolibetpunfto/« erit tempus per/W, «« 
quale teftipori per /i^ , ^^ • 

Hoc enim patet ex pr«cedeniib us « Mam 
propterhyperbolamlinea/Wjaequalis eft 
ipCisf^ yce.pcr$t. tcrti j Conicorum^ . 
3ed rr.fcmiifiseft fpatij liorizontalis d 
h . per hypotefim , ergo asqualia funt tem* 
poralationum tamper /W> quam per/f, ATq^ 
td. Quoderat&c. 

PROPOSITIO X LII. 

SI datumfithorizontalefpatium4^terminatum,&lon<>i. 
tudo alicuius plani/. data fit maior quam d b . Secare o- 

porcetpIanUm/.induaspartesinasquales ealege, vtfiexter- 
, minis ^,^, fadtapIanaadidempunAum componanwr, tem- 

puslationispermaiusplanum,»qualefittempori lationis per 
minus plonum & per horizontalem fimul . 

Hocduplici modo abfoluemitt. Primuai contempladu^, 
fiue per relolunonenb deinde pra^c^ . 

Refoluriuchocmodo.fiiaumiamfit quod faciendumeft. 
&iuuduoplana r^)f^>vtimperatumeft,Qempe«qualia &- 

S mul 





V I S De motm^rMtmJtfiendenr. 

roul ipCi /f ScaSim>^ vr tenipiis 
per #i«>cqualeftrtc;0ipon per>^. 
Is.. Pfodixcarar #^, vtriffiqi in r, . 

#jr |, l/^&lc a^bd. OKjuales ffnt intcr Jc •, 
Cww., Ccrtumeftquia i<^,?iifimula?qua 
lesfuntipfi r,/,pttn<Siim ^ eflein^ 
cUipii , cuius axis maior eft c ^/, & &ci funt ^,f , punda • Cer*^ 
tHQicttiMii^eftxqiii^traQpuspcreajequ^ th^ 

h d , idiem pumSiim e ede in hj^rbola cuius ^bcrikit d^ ^, & 
vcrtex k^ ( diuisa nen^ ak , iaqoatuor partesasquales^ qua« 
tmMTiSL6thb:^hoc2m£m praecedcRci •. 

Eritergp pundum e.in comiminiconcurliidciantm/e^on&t 
fcd dua? fc^oncs datae funt ; quandoquidem dantur foci com- 
munesvtriufq; #, & A > & c d\ data: eft axis maior elttpfe 5 da- 
turq; i h. diamctcr hypcrbolae «ncmpe femilfis ipfius dh \. qua« 
te cciam pundum e ..datum CM^ 

Componetur hoc noodb •. Fa^:^ igitur duabus fe Aibmbufli 
hyperbora,&ellipfirqua?c0ncurraatin c*^^ ^pmfio rducan 
mr ^#, ^f , crunt r ^> #i, fimulaKjQaks ipfi/I & erit tempus 
per rii,aequalctemporipcr r^,^4,,fimul,obhypcrbolam^. 
Quod oportcbat&c. 
. FaciUus tffm% hpc medb pra&ice mfigura. feqventi • 
Secctur ab ipia linea F* pars/»y,qua? ^ 

ae^palis fit kxM ^^j liK)rizontalis a V^ P x ^ h ^ 
^ •. & rcUqua Im «^uidatur bi£uiam ia ^ B 

9., Dtca 41^, «w efleplanaquaKfita, * 

quat^ a puo^ ^ tociisKOW 

piUM^UH»^ ipqitalia iaficnt tcsafiQca Katiqnam^ tampermaius 
planum^Mi^quamperflttniSiit^fimidfcum horisMfitci^»Hoc 
autem perfpicuum cft cx Propofitione 4 1 . cum diffcrcntia lon- 
S^OMitaiifi^^ jfitpfficooAiMaiotKm ab^i^lts icmi(!i 

^atijhorizontalis iT^. Q9od9Mte:> 






pj?a- 



idkmWmml t^ 



I 

DAto horizoniali fpatio ^h » datoq; angulo Ijic » qui mU 
norfitangi^o tKktnguli cqiiiiaceri » (aii^s enitn pro- 
iileg»iiiiblttiiiledEbt) oportet oriaAgatum cooftiniere' qu(kd 
^al^at bafioi ^ ^^ & aneuliim ^ 4^ ^ 
4|aale (« teinporifer f J, ^ ii^ 

Diuidatur ^^^ • in quatuor aequales partes 
.^fUarumyDa&i/^^eic vmice d^ £ocis dy 
fk, ^iiff h)r^i>ola4}iwiecetredam^^,» 
in r . < fecabit enim o^imBQi vt ffifra denuxi-* 
ftrabimus«^ Dico c etfe tertium qua?fiti m- 
angulipundum. Dudkaenim^iwpacetdid^r 
rentiaminter^r^r^^effefenttiremipfius >^ 

propterhyperbolai»,&propicrdiuifionemiineje it^ in qui^ 
tuorxqualespartes^Qiiaretcmpttsperr^^a^qu ' 
per ^ ^^ ^ ^ : Qupd ei^ar &c« 




m^ i !]•' 



■N » 



^odMtfem in pfkceiemi figMrd (Jhpf$Jm dmggU b 
mM$re^itdmfi^f^gfdMsndd0g^liM^0iUt^ acirjw 

ftTboUconuenidi^Jic demonHtabimus infequenAfgmd^ 

Sit4fymfSUoj\>Q.EatgOTi{Ut$gulu99t^ /; 

af ^Msde efi quarff fdriifigmMfer^s. 
tertijConicBrum^ ^l^^ddrdtum etiam fc. 
0fU4iU eSt eidem fudrtd pdattifgitrf^ fer 
frimdmfecundi Conicorum : erunt ergo 4» 
fuMid:mter fe reSdngulum e a f, &qudm jS i :ft P A. 
drdtMmic. ^narevt z(.dd {c.itdfc *> -r «^ 

ddoL ^Jioc eS ddidiStmt igitmfunCtuA ca itofemkircuk cuhts 
didmeter eJidsL,(^ centriib.SedciifintfqndUsh£.z f.fercm 
firuaioneinfrofcjitionefrfcedentii&dngnlidd ijremfini, 

€Bjihzdxis^icdddxemdfflicdtd.eruntnqmdUscb,c^.& 
tridnguUm ^hrc , aquiUterum erit . \^dUht ergo knedqnn 

S a ' ddfun^ 




» i 



CT40 " DemomgnQm^^^ 

4dfun£tum d • augulum Qontinedt cum d a . minmrem Mguh 
SibCftrijinguli squiUt€riiConuenirt0mmn$cum bcj* ^Mfe 
infgurapitjtdtdcntis f^opo^iiohisdined, ic^ cinueniet ctnm 
sfymftoto^ ^ idep etidm cum hyferboU . 

j ■ ^ •• ■ . '. • •• . . ' ■. ; . 

P R O P O S I T I O XLIV, 

EX infinitis fpeciebus tri&ngulorum redangulorum, vn« 
tannim eft qu^ habeathaiic prerogatittatn, quod fcflicet 
tempus per hypotenu fam ^uaie Ot tempori per reliqua duo hr 

tcra. 

£t h(c fpecies-iUa eft <\mx prima omniuni , hoc eft » quae ii^ 
minimis numeris habet tria latera comenfurabitia i Ntmpeia 
Aritmetica proportione numerorum 3. -^. & 5. 

ExponatBr triangulutn iC^ ^ ; cuius latus 
ab horizontale (it 4.&i c. eredum Gt 3« 
4biypotcnuia autem ac.Gi^. Perfpicuum 
ieft anguium aba. re<fhitix efle ; cum qua-** 
dratum ac. 2 5.sequale(itduobus.quadra- 
xis cb^ba.g.SiCi 6. Manifeftum etiam eft 
lempus per r ^ . ^quari tempori ^tx cbyba. 
<umdifferentiainter ac.cb. iita-femiifis fjpatij horizontalis 
ab.opoAtft/^. 

Dico pr^erea nullam aham fpeciem triangulorum redan- 
glilorum habere ilJam proprietatem . Nam ii poflibile efi , ha« 
beat . & iit triangulum iilius (^ccki ipfum adb. 

Quia tempus per 'da • a?quale eft tcmpori per dba. erit di^ 
ierentia inter -a d. di . sequalis femifli horizontalis a ib. Pona- 
turindiredumipfi otb .linea i^^qu^ ^qualisfit femifli hori- 
zontalis ; erunt iam adyde. ^quales inter fe ; & ^/^ » r ^ , ob ea- 
dem cauia squales inter fe ; quod impoflibile eft . ImSta enim 
ae. eflet vterq; zngudusdacy cacy rauaHs angulo e . quod eft 
abfurdum. Nullaergo fpecies trianguloruredangulorum rcpe 
ritur^prneriamdi^tam qu^ habeat fuperius eaarratam pro- 
prieta&em. 








fofemms etim demnBnire ex imfiMtis /peeietMs tridngm^ 
letMm ohUqMMHgMlmrttm > 'qMM vnMm MngMlmm dstMm hdbeMt^ 
fMtd 40 . gr^dMMm ^ VHdm tdntMm/feciem efeqfutprddi^dm 
prefrietdtem hdbedt . ^inetidm e^eiideretMr ex ittfinitis by. 
ferbeldrMmffeciebMs^vndmtdntMmfpeciem tfieqMd iUdmbd^ 
bedtfrdregdtiMdm . Sedttoneftumti emnid hsc minntd emt^ 
cledtimfercenfae^ vtleffmsfdtientid^benetuientidq; vlte^ 
wiMs dbMtdmnr . 

P R O f O S i r.IO ZLV. 

« ■ 

SI Fuerit quodcunq; mangulum dbc^hz 
benslatera db^bc inaequaiia» puta db 
maiuS) b c minuK & bafim horizontalem . Di- 
co eodcm tempore fieri lationem perbA. fo- , 
lam, & per ^r , fimul cum tanto* horizontali -^ F I D <J 
^tio quanta cft t>is difierentia inter ipfa latera . 
Sitenimdifierenriaimerlatera cdy cutusduplaponatvr ce. 
PeWpicuumeft b r,r^,fimulacquariipfi bd. lam fifuppona* 
mus tempus per ^ ^, cffc bd^ erit tempus per bc. jpfa bc^8c 
poft cafum bc , cempus per r /« eritdimidia ipfius rr, hoc^ft 
c d. Aequale eft igitur tempus per b ii temporiper bc^cd^ 
mul. Quoderat&c. 

I jfdem pofitis .• quando ^ r ^in eadem figura ) minor fiierit 
quam bafis trjanguli • Dicoduograuia eodcm temporis mo* 
^ menio demiflTa ex b per latera bdybc poft eonuerfionem ho- 
mcntalcm fadamin dy&, ^«conucnire in pu^o bcfisyHquod 
quidembi&riamfecetipifam de. Oftenfum enini cft eodem 
V mpore peruenire duo grauia ad punda 4. & ^. ergo cciam re-* 
liqua fpatia df^ ef srquali tempore peragentur,cum fint aequa- 
liaperhypothefimV&gnidus velocitatis acqua^^s ^nc.per Y«. 
huius» 



t>RQ. 



14^ Lemimft^km JUfciHdent, 

F * ^ f O S l T t m ^LVK 

POiito qnoiibec tria agitb M^€^<sm 
bafef^honKOBtiparaUeiaiik^ St ' 

graue ex^ece ki veidce 4 pcratofinaa A. , .,, ,° 
latasi»fcaqWtiodej»er!Mfi«*A^uBi :■- / \ / 
itnpctu coDcepto conuertatur, bafiq; per / \/ 
z&z cum eodero impetu per alterum li- 3 ^ 
tus A ^ afcendat, impetas ille pffcbicet 
graue pcr aibenfum i ^ . vfqiie ad idem 
pun^um^.exquodifccflfarat. . . - ' 

Compleaturpandlelogramum .**.cii^*«itq} ^rf* lionrou- 
-talis, & quia impews aquifituspcrdcfcenlum ^*. perdttcit gra- 
ueperpranum^</.vfq;ad</. pcrScholiura Prop. ai..Galilei 
dc Motu Accelerafo, iaeminjp<$tas <poft wanttiito bafim 
motu«quabili)perJucetmobilcex ^yk^tUi ^.liwxcaim h^ 
.tf^/jequalcSj&aBqualiicrinclinatae^ Q^e4a:« 

Lemma. . . 
Silnter fiitdUUshmamtdes a b, c d. 
dit^Jinedfufriut bd^daL.eritttti^s ^m- , 
fusfervMMm b d , ddtemfHs afcenfus fet 
Mterdmdsit vtefiiffa bd i«/ da." EB' 
tnim temfus edfus per tfM^meumqv Iiiretm$ 
exG^dileOy^UAle ten^tri 4fe*»fus fer ean 
demyquMidefiMafeet^ustumimptmfetdefctnfum 4qtufii9* 
Sedtemperdeafuum fer bd, tf ad/iMr/ v/ b d Adxdt erg§ 
etidmtemfuseafusfer hA^Adeemfds^^eenftu fer da» «'// 
^bd.-^da. ^spd&c. 




P 



IJiOPOSJTIO XLVIl. 

« 

Ofito quolibettriangulo Jthc. cuius bafis b c horizontalis 

fit , fi fiant lationes ex quiete in vertice d . vtrinq; pcr tria 

' latcra, • 




Lihr Prmm. M4i 

latc» y erit tempus lationiim per 4 r » r ^> S^j^ «-• 
^pidleceaiporilatiomuxiper 4i, bcyHd^. 

Ponamusenimtempuscaiusper ^^«.efTeipsi 
i^^^erittempusLper ^r.feioiffis ipfius. bc^ per 
Propofit..4o«iiuius»cum ^^fithoFizomalis/traa. 
imifla vera bafi motu aequabiii », tempus ofcenftis* 
pcr ir iceritipfa € d pec lem»^praec6d.£ockm^9d6: cum fit te 
pus caiiis per di. ipfa.4£^erit tepus cafusper diCi^^dt^Sc pef 
horizontalem e k eriticmiifis ipfius c ^, indcfier afcenfum h d 
erit b d. per lemma prascedens«Eflergp tepus per vrramq>via 
tamquam duo lateratrianguli fimul cum dimidia bafi • Quare 
tcmpora per vtramqj viam> fiue d-JkCdi fiue ^1^:^41^ aK]UdUa ii;i«^ 
tecfeerunt.. Quod&c. 

PoJlifitJimiUddfmiM^dnd^fgimtfcI^ irreguld^ 

rsBMJi, Sed cumhfcommdjplirdbreuudtc cxequi nonfojjtnt^ 
cxiJimduieorumdemon^TdfiancmiJ^ud crudiias plus. mtlefHM 
dUdturdm^qudmdoBrind.. 

■ 

E R OF a s i T £ a^ xirijj.. 

AD aliquod perpendicduin:^ta.dtiO}plan}i diueriR IbiK 
gitudinis ab eodem horiaontis puDifto wA^^i^ uuvt 
temporapefin£»apiisuD2^«fquaiiaiiiit« Vdi« 

Proponir aliquis geminos afleres 4>&iu. 
diucf ff longtfimtiais, ealcge vt ab vno eor 
dcmqspumSb iapaaimcmo infle^ti dcbeaar 
ad panetcm>& grauca cx £iiiigiis eor2. eo» 
dem tempocc demi£i , fimul iu^t:mq,\ush^ 
pore &raBiuff in iQrram *- 

Compooatitur djic k ad,aogul(brc<ftuflK: 
fintq ; cdydCjSc produdia c Cy ^ pcrpeii* 
dicttMsfit^^/. Accipiariiriamifi pauimowo difta^ 
paritte^/>qua?arquaIisfitipfi/"^;Tumapun£lo 6. iaA^^tanr 
vitcsiipi^kMm^JkiJd^^^jj^^ 9dyt dcv 

DIlo. 




J44 ^DtmomQrauiimdefcendenL 

Dico tcmpora pcr / A , & per ih . asqualia efle . 

Concipiamus bafim c e . qrianguli e de y clTc ad horizontem 
eredam . Manifeftum eft tcmpora lationum per e d^ dt. ^qua- 
lia eflfe . per lemnu Propot 13. Sed cum duo iatera c ^, df. 
duobuslateribus ih. /&^.a;qualia (iiit vtrumq; vtrique,&an- 
guii cfdy ig h . itQXy fi ex quadratis ^quali($us dchi. deman* 
turquadrataaequalia ^Z*. ^^ . remanj^buat o^quaiia qqadraca^ 
f^ yg ^ > & ideo/5r .gi. lineae cquaieis erunt i & propterea inte- 
gra triangula cdf.ihg. jequalia,&fiiniliacrunt.&tcmpu$ 
per ih • aequaie tempori per cd. 

Eodem modo oftendctur tempus pcr7i&. aequale tempori 
er dc. C^arecumxqualiafiattempora per cd.de. aequA* 
ia erunt etiam per ih.hl. Quod erat &c. 

PROPOSiTlO I L. 



\ 




SI ex ^ piUK^o fublimiori cir- 
culi ad horizontem eredti 
traue cadat vfque in centrum b^ 
: inde per quodcumq; planum 
iiue eleuatum, fiue deciiue, con- 

uertatio: cum impetu iam conce- 
pco;grauehuiufmoditempoFecafus 4^ abfoluetfpatium hdi 

quod nempe aequaie fit vtcifq. tum femidiametro b c % tum eda 
ipfius perpendiculo ce. 

Seeuutcdpiuaiisip&ce. Dicptempusper^^,exquie« 
cein ^,^ualeeuetemporiper bd. poftcafum ^^•Eft.n.ob 
^ualitatemvt dc ad cb^ ita ce ad bM^ hoceft cfflAfb^St 
permutando vt ^^ ad cf. ita cbz^A bf.in vtraq. fi|ura . 

Sedinprimatantumeritcomponendovt^//ad c/y ita cf 
ad hf. In fecunda vero erit . Conuertendo , per conuerfione 
rationis,&iterumconuertendo,yt 4f/ad r/, ita r/ad hf 
Qjmre in vtroq. cafu tres linef df cf bf. funt in contmud pro« 
portione . 

lamfitempusper ^^ponaturd& 4h:% erit^cmpusper/^ip 

fa/i^. 




fft/i(,^per/^tmpi]seritEDediapropomonafis/^ Qtiare 
tempqs per reliqimm Unedb « i»empe per t d^ cfk reliquum tem« 
pori$,nempei^r. Idem ergo tempus eft iationis per di ck 
quietein 4>&per ^^ pQftcafum ^^ • 

InbM jrof^fiMne reipfk demonfifdMHft dw Theottmdtd 
GdUUi > HeMetu dccelerdio \fed quid^dlde ddremnoBfdmfd^ 
ciunt^ eddemdinerfd iternm rdtiene c$ntemftdb$mur > vt Incem 
feqnentiCeroUdriefrdferdnt. ' 

Sigrdtiendt$trdUtercdddtex a in b. 
{^exb ttim imfetn conceft4h^per qmod^ 
Ubetfldnmm b c connertdtnr . ^adrir 
tttr qudntnmffdtif fer fldntnn bc dtjol 
Udt mobile temfore cdfus a b • 

Fidt circddidmetrum ab. circulus 
a d b , centroq; b , & interudllo b a r/r- 
^i^iW/ ac • Dico^due deUffumfer a b 1^7^ exfumio b rjfffi 
imfetu concefto conuertdtur fer fUnum inclindtum b c> /r^/^ 
rr dqudUtemfori cdfus^fercurrereffatium dqudlevtrifqi fimod 
bc,cd* 

Si enimfoficdfum zhgrdueconuertereturferfUnum qnod^ 
cunq; b c > moruq; dqudbiU frocederee^ grdUe huiuftmdi fer fU ou M, 
num h c temfore aqudli temfori cafusffdtium fefdgeret dufln 
iffius b a 9 ergo fercurreret ffdtiumdsifUmiffius hc. temfore 
Cdfus ^fifoii cafum dqudbiU motu f/rocederet , Sedfuferneniem 
teoferdtionegrauitatis^obilenonfrocedet motn dqudbiUfu^ 
ferfUno b c i Sluin immo tempore cdfus a b ^grduitds fromouo 
bit mobile fufer fldno b c tdntumffdtium qudntd eSt incUsd im - 
circuloUned d b (quo enim temfore grduitds trdhit mobile ex 
a iu h.eodemtemforetrdhitetidmex d in h fer if.GdUUide 
Aiotu Accelerdto .)Ergo dufU h c infrimdfigurd dddendd erit 
4 b » conffifdnt enimdeorfnmtdm nootns dqsutbiUs^ qudm motus 

fduitdtis i Infecurtdd verofigntdi dttfU h c detrMtendd erit 
h(quidmotusgrduitdtiscontrmttsefiimoinisqudhiU)^ 

T coft^ 



/4< ^mtu^f^miiJi^^ 

CorpUariuiii» 

fr^ CmlUrk dmimddMniimas qnd 

fiffdm diifmdtxfMm&0 a dmfelU^ 

tffrm$f0 veldfij^tm fjtrimrimmtdl^ 

aJb ^f€r4gMqiSi»t0 Mi§ti$$€9ffmt€ffM 

tium zbiMqueeodtmttmftTe qtt^pf 

Utie zh gtdHitds $9MU ndtftrdU dcar^ 

fttm trdhdi fer rdtttttmffdt^qMd^d ep 

a c • ficettttet a ittterttdlk e b fttt eirctdtte b Aei ^•&cifrcddiM^ 

tttetrttmzQdUtitcirctiUt.zhiCyWteitdeitf^tdftmttAe^tlem/im 

fer imfetuferfUttd ^ d >a e> a f » e$demtett^eferdgetfittfft^ 
Ut ittterceftds c{yic yhdrZh^cetitmtit^d/tttritMtttmm 




<.« 



SI doo^raunt lieiiuKantar eodeia «coipons moincMo^ 
^liei6pianiekiutipuoSis , & poft cartnn per eandcfii 
'horizontakm lineam conliertaatur ; grauia in quodanipuo^ 
Imul coaiiffifentt^qiwrf in iKNrizoataiicantumdiftat a ^onoe- 
kiHUO'^(uaiaa.cilduplan)edi(propoiuonaIi» iotcr al t it ndinr t 
diiium» 

Sitplamun deuanBn s^, tnquo fimu»- 
mr dupi^iaslibctpundbi^ ^ .» <3piibii»duo 

frauiadeniitraotureo4eiBiimnl<ten^>ore* 
ita)9cm>4rinediainter4^,^r.&ipfiu» 
4 f . dtlHa fit hocizoBtalis k i, Dico^uia 
-cptLi^rbpuiido dcnifiiex 4.8cr.inpaiido/comMtiire. 
iungaRireium «^*&conipleanr parailcfo^mraum ^tfg* 
CUiiK|;fitii^«<dteplaipfiiK^i«»crit/r.lioeeft^l.duirfa rr* 
laiQ fic* Mobiiepoft cafiun r i . ftio impetu currit liomonta« 
Jitertempoiecafiisri^di^iam r^. tt^tempore rr.curr^ 

^IMkniflfettdupliiBff « hOG«ft ipfiun l^* Ttmpote ^ctar 

iitte*. 




Liier Pfmmi 14^ 

mtegroe^ fiuotlatfocicsper c^,& Sgfictodenektempon ftt 
ca(bper^irfquareeodece{torkmonKntoeitmtgra afcer& 

3iiidem in g ; alteruni autem ia h . Sed reliqua etiam ipatia ^ 
«^^•fqualibustemporibus peraguntur. ^Velocitates eniitt 
funtvttemporacafuum,hoceftvtr^.ady^.fed fparia iJ^d 
ob fimilinidinem trianguloruiti (unt vt veiocitaces^quare vti di« 
^kiiqfMft ^ualibustempodbus peragcntur «^ 

Sunt ergo coniun^flimtempora pct di^^J. ^ualia tempo^ 
ribus coniun^im per chybd. Quare duo grauia&c» conue* 
niefminpun^i/t Qaoderac&c. 

I^em^iterdemonftrabimus , 

&$mftts vtc$t9q^ dtituiinibHs ^c bc. Demittdnttir dtt4 
^gtMid e^dfmtemfort ex tiydrh. Sit^; d c. medid intcr^dftis^ 
cuins dnfU fon4fnr h$fix»dntklii c e« 
Dica temforn tdtionnm ace.bce. ^^ 
^itdtidejfe^ 

JFidt circd didmetrttm ca fdrdbold 
ftt^nn^ne ^l^ndvihrticem hdbedU in c. -^ 
dncdntnrf, ordindtm a f . b h . NotnrA ' 

tfimfdTdbotdttdfffe zi.dd bh. vteft 
ac dd cdfVelvt cd^ dd cb. 

Idm. Temfttsferzc.efiz£^fjrfer cc.foIicJfnhtic.eJl 
b h • {fi enim temfore f a .grdne^cnrrit dufldm a c, temfori b h, 
tmrretdnfldmdchocefiiffdmcc cn^ fint frofortioniles f 
a ddzc^vtlih^dddc.JEodemmodo. Temfnsferhc. efi 
• hh^drfercc.foficiffnmhc^efi^if .( fi .n. temforehh.onrrii 
dnfUm h c » temfoire a i .ctirret dufldm d c. hoc efi iffdm c e . 
c^md fitnt fr^firtiotodtcs vth h .ddh c . itdSL fdd dc .) 
£rgo temf^ Idtioknm a c e .fnnt tinen a f . b h . Timford dsi^ 
oemtdtiottitm hct.fnttttinedhhyzf.^dreconinnitimtem^ 
f^rdfer ^C^cc. dqtkdiidfnnt temforibns fcr b c • c e. cottintor 
&im. ^odetdi&c. 

. CoroUdNnm Primttm . 
JHUnc maniiefttim eft datoquolibet fpatio horizontali ir. 

T 2 cuius 




1 48r De motu grauium defcendent. 

cuiusfubduplaponatur ri/« Sicircamedil 

r ^ » d uae.in continua propordone fumantur ^ 

fe^cb* Tcmporapcripfas.rf,r^,aKjua* ' li 

liacflctemporibuspcr bc^cd. j^ ^ 



CoroUdritm //• 

Manifldum ctiam eft tempora perpendicularium, & tcmp6 
ra hbrizbntalium lationum reciproce awjualia efle . 

NajB in figura vl^im^ denionftrationis , tempus perpendicu 
Hac.tik af^ eademq; af. eft tempus horizontis c e . poft aU- 
umcafum^c. 

Tempus autem b h. eft tempus cafus b c. idem vero tempus 
eft horizpntaiis latioiiis poft aliutii cafum 4 f , 

PROPOSJTIO LI. 

SI fuerint duo plana ^qualiter inclinata> a b • maius, c d. rsA^ 
n\}Sydcb J.tn horizon • Sumaturq', 6 1 . media propor- 
lionalis intcr longitudines planarum j & du^aa t cf^ ponatur/* 
^^duplaipfius ^^.Dicograuiacodemtemporedemiflkex d% 
Hc c. poft cafus ^btcd^ixx pundo g . conuenire • 

Sunt eptmper praecedentem tempo- 
rabtionum per ^b^fg. fimul equalia 
temporibus per c d . Scfg . fimul . Scd 
etiamtempusper bf^ poft cafum ab 
aquaturtemporiper df poftcafumr 
^•^cumfintfpatia bf. df vtveloci- 
tates ebf cd.J. crgo coniungendo tempus pcr omnes db^ 
bffg. «quale erit tempori per omnes cd^dfyfg^Scii^ 
grauia conuenient in g. Quoderat&c. 




s 



PRVPOSITIO LII. 

I Fuerint duo plana ab. maius, cd. minus, aequati- 
ter » inclinata^ ita vt lationes horizontales 9 polt ca- 

fus 




Liker^rlmus. 149 

fus in prima figura contrari^ inuicem fintjin fecunda vero ad 
eafdem partes • SuniatiBra; media proportionalis inter longitu- 
dinespianorum if^Scht J/dvph gCj hoc eft ditferentig 
incermediam bej &minusplanum>dudadeindeyi&)a?qua-- 
liipfi r^,&paralle]aad ^^iiuDgatur eh^ quasfeccchorizon 
temin jf ^ 

Dico gra 
uia ex 4.& 
c • eodem t5 
pore demif- 
fa, fiverfus 
i conuertan 
tur in pudo 
i conueni- 
re* 

Ponamustompusper cd.tStcd.ytlbg. {Ibt (equalem # 
Ergoinhotizontegraue r.tempore tg. curret duplam bgy ' 
&tempore^^.curretduplam jr, ncmpe df. £ftitaq;tota 
eb. tempUs pcr c df. Eadem quoqi tb . tempus eft pcr db. 
Quar e codem icmporc peragenmr c df. Sc db., Rcl iau* au- 
tem fi.hi. eodcm temporc peraguntur fcwm propttr nmiliru- 
dincmtriailguloiiimfpatfa/i)^/» fintvt vclocitates hf^ebJ) 
ergo coniundtim idem tempus crit tam per c diy quam per 4 bi. 
Quaregrauiaconuenientin i. Qupdcrat&c. 

In fecnndafgwd non dehent Utioncs horizantaUs ejji con-^ 
traridxndm graniantinqMdm conttenirentfed ambp wirfusfar^ 
tes-i. 

, y" p R o p o s ir I iiii^ 

DAtis duobus perpendiculis abjcby inuenirc fpatium ho- 
rizontalcquod cumalierutro datorum perpcndiculorii 
codem temporc conficiatur » 

Ponamr bd.xqixalis ipfi ^r,&circa ad. fiarfcmicirculus: 
ponaturq; horizontalis be. di^laipfius bf. Dicolationcs 4 

bty 



<t« 




iio DemmGr^tutmdefctHiemi 

biycie. c6dem tempore abfoiui« 

Hoc eniiri patet f>€r CoroUaviufn prim& 
Fropofitionts $ i. NamaldtiidinGs perpen^ 
diculares ^^«^r.funtcontinuji^propoiffo 
nales circa if. feinKIem fpattj hiwiMntaiiA 
Quare h6am eft quod &c. 

pRorosiTio Lir. 

• * * 

DAtoquoIibetperpcndiculo,& quolibecfpario borfzoo- 
tali ,• aliud perpendiculum reperirc » quod cum dato fpa- 
tio horizontali cod^ra tempore conficiapur acprimumpetpea* 
diculum cum doto horizonte • 

Sit pcrpendiculum damm ah.Sc ho- 
tvton 6 c . cuius femifns fit i d. lunga* 
tur ^k/ifiatq;angulus tdf.^A horizontfi 
a^UatisanguIo bady qjlkieft ad pet^n* 
diculum^ Dico tempora lacionum per 
fb .S c iimaU &pcr ^ ty h r» (imul, aequa 
lia eflfe. Triangula enim redangula /* 
b d^ddh. fada£jnt a^qwangula . Qua* 
ttvt/i^.ad^^.itai^.ad ^^.Ercum^^.remiflls horizdn* 
caiis fpati) mediafit pli^rtionalisinter perpendicula ft.di: 
crunttcmporapery3Jr,&per^iv.a6qoalia. Quoderaf&c; 

P R O P O S I T t Z F, 




4s«a"* i3 ipias tf^^.poftcarum<4brcuioritemparepercttrri,qua 
aliud qiiodcanq;perpenidiGUliid]f cum<eodem ^abo horizoiKa* 



SI itKkhotitohtxlis j$. dbpta ^f|>eHdi^ 41 1, Dico 
ipii 



Erigatur ^ ^. perpendicuiaris ad 4^ jt 
Scper tfii pun^, circa diametrum rd ^ 
a^turparabola,^^i.cuius£x:us«rit ^ 




/ 



#«(poi$Cii<Bimeft rfi.duplafflfiiis4<..) Suniatttriunqiiod» 
libetalJudperpeiidiculuin ri^aducaturhorlKomalis id/. 

Tempusper r^eft ^^.tempusautemper ^t. eftidem ac' 
tempuscafusyergocempusperr^^.eftipfa 4/^ bisfumpta* 
Sedtempusper ri.eft vi^ctempusaucemper i^/.quantumfit» 
fic venabimur • Veiodtate d b . tempore d i. currinir 4 b . Sed 
velockate r i . tempore di . non curretur eadem ^ /^^ Fiat igi^ 
tur vt velocitas r / • ad velocitatem ^i^ ita tempus d i ^^d aliud 
ml. Et critm /.tempus ipfius 4i • poft cafum ri .. PateiLergo m 
Lci. primam & tertiam-proportiojiaiium , «aioresefle quam 
dupla medise » faoc eft quam di^ bis lunip^ . Quar e &c. 

fjtopeijTio Lvi. 

SI di. korizontalis dup 
laftieriteleuat^ dt^ Di* 
co , quo loQgius a pun^o e. & 

demictatur graue , cotardiu& ^ J 

lationcmfuamvf^;iiii^abibl. c .^^~v l 

uere* __^^^ "^ 

Demittatur ex pun^s r, & 3 
d. Duo grauia; oftendrn^ 

dum eft niaiori tempore fieri larionem per i/-c^.qiiam per ^ ^ 
i. Fiat circadiamemim 4ir,|)arafo9la i#i»/.&ipfis cM.^e. 
fittertiaproportionalis d^. Ipfis^iutem i/^,4r .tertiafit 4*, 
& ducantor ordinatim linea? ex pun^ V) r j^ ^ ^ ». 

Quiaquadratoeidem^r, asqualeeHvttvmq; rcaangtrhm) 
^rfjJ.r^jf *enm{h{ceademrcdangula«titiali4lhT0l' fe; pro- 
pterea latera reciproccproponioiudiaiiaba*ftt vAempe vt i^' 
ad ^r.ita r 4,ad dd. Sed inhaccademproportione obpa«» 
rabolam^fumquadratar^/lad^iir^A r ?ad V^ , ergo propor 
tionaliafunt etiam latera,nempe vt hf. ^dgm.ixzcIzA di . 
extrenurautem ^M^i^r maioresfoMquafbiHedffc^, cl^iSc^ » 
estrem«fimuifumtempusl»fonis ddi, at^iic^^fanttempus "^^-^' 
Jationis idi. QiMu-etaidiu»|AfolttaBrljifi9^ 4uam.Sir. 
per fdS. Qsioderat&c- ^ i-r» , 

Idem 




i^i Demomgrdwumdefeendknu 

Idcm infmur etiam de piitiftis^»&>&»fupra ip/bm €i ifiunptif» 
Sunt cnim tempora eorum > aK|ualia tmportbus |mndloriiro e. 
xnde ean & ^.vtrufnquevtriq;&c. 

dem , 

PROPOSITIO LFII^ 

^ m t 

Sl ab aliquo pun<fto linea? circulttm tangentis in pun^ Ah 
blimi, grauia cadant in periphieriam Sc inde per chordas 
horizontales conuertantur • erunt tempora lationum per vtram 
que chordam S^ ei us perpendiculum , asqualia . " 

Tangatlinea ii^,cirbuIumereAum, in pundo fublimi ^. 

Et fic tangens horizontalis omnino erit / SOmpto dcinde quo- 

libetpundo ^. grauia demittantur perpendiciUariter in peri- 

. phaeriami & conucrtantur fiue in r , fiue in d . Dico tempus pcr 

i^r^.&per iii^.idcmeflc. 

Sunt enim horizontales c e > df. aEX]ua« 
les , cum c h . fitparallelogrammum reda 
;ulum , & r^ y df. iedae fint bifariam in g 
lc^&.pundtis. 

Quia ergo ^i^.femiflisfpatijliorizonta 
lis media proportionalis eft inter altimdi* 
iies perpendiculares sc^dd. ^linea enim 
rfiJ .tangit,& 44^.circulumfecat)eruntper CoroUarium pri- 
mumPropofitionis5o.huius,temporaIationum dce^ ddf. 
asqualia. Qtioderat&c. 

Et per fecunaum eiufdem Propofitionis Corollarium ea- 
dem tempora reciproc^ eoualia funt &c. 




P R P O S I T I O Lrill. 

TEmpusperaxemparaboIa?» & eius ordinatim applica* 
tam fimul , aequale eft tempori per quartam lateris Kdi 
partemt & eandem ordinatim applicatam « 




LskrPrhfif&2 

Sitaxisparabolx> ditcmordinaximzp 
l^cata^/. £t&cetur^</. fleqikli^quartx 
partilaterisredi. Dico tcmporaper<<^/* 
6cp€T Jif. aequalia effe inter fe . Diuida- 
cur ir.bifariamiri r,: 

Erumtumquadratum «jjtumrfdiangti- 
lum <c^(/,fubquadnipUquadrati r^> Sunc 
idcoxquaIiainterre>&ipra «^.mediaproportionalu eftitw' 
ter <<^,^i/.C^areperCorollariumprimumPropofitionis f o< 
huius , tempora per dic.Sc per J& e xqualia funt &c. 

Suntetiam per fecundum elufdem Propofitionis CoroUx* 
riiun, reciproc^xqualia .Qupd fatis fit ofteadiirecircaaxxuai 
grauium nacuraliter defcendentium . 

FmhPrimiLihi* " " 



K D£ MOJ 



DE MOTV 

Proieflpruin • 
LIBE-H SECFNDVS . 

ROIECT A tmnc yhellortm^immdtyMqut 

> iirc%t(0 terHteHtid ditemHs : $HfremMs hic U' 

; hrnm GdlHeifruitu/ , /u^rgmd etiim^lwidt 

Oftendif GdiileMj inl^rp 

4e MotM Preie^orum,qu0d ^ 

ft molfile dliqtiod a fldno y^""^ A 

horizontdli ab deciddt yimpetu frihs hiri- C ' 

xantdUth concefto ,fdrakolam dUqudm ,vt * 

. b c . cdfufuo defigndbit , Verum efl ; dummodo Uned a b qu^ 

eftdireliiofroitStioms ddh0rix,ontemfueritfdr4Ueld, dr qudn' 

dofordboUittitium \i ■,fa6tumfuerit ex 'vertice fufreme ifffut 

purdhoUjftuef^ueitidem efi) ah extremo dxis faraboUci fUtO' 

.£fo b . ^dttdo vero UuedfroieSionis a b non hori\eataUs,fed 

furfumfutrit , vtldeetfum iaeUndt4, erit quidem Unedfroieffi 

qudddmcurnajdrftfccontingentinuieem tumUnea reHddi- 

reBionisiuxti qudmfaSfafieritfreieSiioy tnm curuaquaerit 

femitafroieiii ; eirftinRum contaStus erit idem acfunBumfe- 

foratienis iffius froiedi ah inftrumento imfellente . Sed 

hancUnedmcuruam ^ ejfe faraholam , ^ edndemfrorfusfard- 

bolam efte , qud di eodem mobiU horizoatdUter frius concitdt» 

exifftusforaboUt vcrttce defcriberetur , hdclenus deftderdtmt 

fnagis , quamfrebdtttr . EBfrofe^o eademfaraboU , vtUtt if- 

feCaUUtua^rmdtin CoreUarie Profof. y.de motit Preie^orut 

neq\v€riftmiUerdtddeoocuUtuiningeniumnehbenefriitscir- 

fmtnffeffdfoft^iffe . Mtamen , veritas ilUus CerolUrij manife-' 

SdfeaitMsnsnmtilUSf^tiibMsobUqiiitdtesfin/^olarMmigno- 



Uler Sehmdui i isi 

minsfammms mnfuetit . Cmmitaq\ froitCliomisvtfUiitimsmi 
famtfer limeas ad hoYix^ntem in^Unatasy ex qnibns orimmtttrfd 
tabola obUqna^non habentes initinm ex vertice^ qnalesfreqne^ 
tifjime occnrrnnt in omnibnsfere iaitibns machinarnm , imm^ 
etiam neq\ verticem, neq\ axemhabentes^ qnales fnnt froieSHo 
mesimclimatfdeorfnm > Incem CoroUario Calilei^erre conabi^ 
mnr^ & cttinfmodifit limea cnrnafroieiiornm vmiuerfalins dc^ 
termimabimns . 

Defimitio^ 
Diredio proi edtionis dicitur linea reda qu^ cangit lineaiiu 
curuam proiei^ in primo pun<5to eiufdem lineas cuniae. quas 
quidem dire<5tio in tormentis bellicis efteademac ipiius ma-^ 
chinasaxis .. 

PRaPOSlTIO f R I U A. 



SI graue furfum proiedum ex d. aficendat motu 
Deficiente vfq; ad fublimiuspundum fuaslationis^. idem 
vero mobile a?quali tempore ^ & eadem velocitate quam in pu« 
^o a habebat,fedmom»juabiliafcendat vfq;in r# Dico 4 
^» dupiam effe ipfius ah. 



MiUtk 

SAMo 



Si enim non eft dupla , ponamus aliquam a d dup- ^d 
Lunelfeipfiusii^. ^ 

Concipiamus iam cadere naturaliter mobile ex h 
in a. Gradusilieimpetusacquifitipoftcafumex ^in 
d. eft ille prorfus qui vehit mobilcad eandem altitudi t 
nem h. eodem tempore » & motu naturallter defici&e. - a 
Idem verd gradus i mpetus eodem temporei fed mo-* *^'' *^ 

tuasquabiliperducitmobile ad altitudinem ^^duplam cafus 
h a . Std ille idem impetus qui per fiuppofitionem perducit mo- 
bileex ditib mom naturaMter dt^ficteme, illud perducebat 
edammotu^uabilieodemqjtempore ex ^ in ir. Vnus erg6 

y % idemq« 







^iS4 ^e mofu Proie^omm 

ideinq;gradus impetus eodem cempore ; motuq^* squabili per« 
ducit mobile per duo fpacia inasquaiia 4tpj$J^ Qiipd eft ab« 
fiirdum« 

PROPOSITIO II, 

SEmita proiedorum^qufcunq; illa fiti^fublimiori fui punAo 
bifariam fecat perpendicuium quod inter tiorizontem» Sc 
lineam dire^onis intercipitur « 

Proiciatur mobile ex d iuxta diredione 
vtcumq ; eleuatam d t . Patet quod iine tra- ^^ 
^one grauitatis procederet mobiie motii *^!f^^ 

tedo 9 &^aequabili per lineam diredionis 4 £ i -J^ 

h. Sedgrauitateintusoperanteabipfadi- 
re^ione ftatim declinare incipiet > crefcen- 
te femper deniationis ineniura;& defaibetaliquam lineam cur 
uam Acd. qua^cunq; fit. Hasc linea pundum aiiquod fublimius 
ca^tens habet ; illud nempe quod efi aicenfionis extremuni > & 
primum defcenficmis • Sit huiufmodi pundum r , & per r du» 
caturperpendiculum bch. Dico hb. duplam effeipfius hc. 
. Abilralumus motum horizontaiem;hic enim m6tus,quo ad 
lationem perpendicularem de quaagemuseft tamquam non ef 
fet ; cum iUam neq^ iuuet « neq^ impediat « Concipiamus etiam 
mobile liabere ien^er fecom (iium perpendiculum hi • hori- 
zontali quadam latione vna cum ipfo translatum ex d verfus ki 
in quo perpendicuk>afcend«tgraue , motu quodam continuo ^ 
fedfempermagisacmagisdencientc^apuado h vlq;adpUR« 
^um^r. Con&:it€tgo mobile in fuo perpendicuio tempore 
Exempli gratia •€ . ipatium Jfc. {cd& mocu asquabili afcendif» 
fet cum impetu & cempoce oodem^ reperketur in h ^deberet 
cnimobmotmnaequabilem effefcmper in communi fedioae 
JUnearum 4^,^^./Qiuref erprascedentemi fjpaducn hh* ip« 
fios hg dupium«il«Q|jiodcrat&;c. 

PRO. 



LiSer SecimdMS • 



'57 



K O 




P R O P O S^-f T I O 11/, 

t 

LIneacurua,qu«dcicribitur^mobili fecundum quamlibec 
cieuationem proie^o > parabola eft , & prorfus eadem , 
quamdcfcriberetmobileficum horizontali impctu proicere* 
tur a vertice eiuidem linea? curuae • 

Sitlineaproicdioni$dire<5Uua 4^. vt- 
cumq;-elcuata > & linea curua 4cJ eycu- 
ius fublimius pun^ffaim fit/. Diicamr po-* 
pcndiculum l/g.Sc ctunt per pnecedcn- 
da aequales //.y^ • Ducamr horizonta^ 
lis/i , & perpcndicularis 4 A^ ^runt iteru 
dKjualcs //. sAy & wi, /^. Diuidamr d 
k . in quotcunq; panes ^uales ir /, /i, / m 
mh.Sc agantur perpendiculares per pun- 
Cttkl^iym. Mani&Aum eil quod fpi^a asqualia ^l, // , imym 
^,percurrerenturamofailitemponbus asqualibus, fimom a^- 
quabili » & (ine acceflu noui moms deorfum y ab imerna graui-- 
cate procedentis, moueremn Sed<]uia ei fiatim atqi aproicien-» 
te dimittitur in 4 . fuperaduenit attradio grauitatis, incipiet co« 
dnuo d Hnea direcftionis deorfum deuiare , & dcuiationes tales 
erunt vt linea l c . defcenfus vnius temporis (it vt vnum : line^ 
vero /^.defcenfus duorum temporumiitvtquamor , & me. 
triumtemporumvtnouem» hf. quatuor tempomm vtf 5m& '^CfA 
fic dcinceps ea lege vt femper<)eicenfuum fpa<ja iim vttempo 
rum quadrata . jQuia ver6 4lyliyimy mS , funt asquales , eruitt 
b<0.^iysny nf. ( quod inter eaiidem parallelas iiht) aK]uaIes .« 
& cum Gxtfyid^^ crit m 0.S . €T^o reliqua need vnum. fquan- 
doquidem tota m e erat ^. ) Ipfa verq iJ . poiita fiierat vt ^.nec 
immutatur • # / aurem ^qualis ipfi mn.cfis.8c addita / e , quas 
pofita ftierat viium> erit tota # e ytj.At h 4. ^ualis ipii hfcvSL 
^^.£rg6cumfpatia/if:i,/rf^i#,tfi&.iinta?quaHai&>2rr,i^^r9 
^^•iintvt i»4,9.i6.&iic deinceps vt reliqui femper huitae^ 
riquadraci»erit linea procedens ex/lper pun^ e.d.^^4 .pa- 

rabola 



/ , 



i$s He m0tm ProieBdrum 

rabola reda cuius vcnex/. & de qua agit Galileus . Scd h^c 
eadem linea eft tradus proie(5)ioiusoblK|U^ ex ^. fad^ per fup 
pofitionem noftram •Ergo linea curua>quf deiciibitur a mobi- 
li fecundum quamlibet eleuationem proiedo , eadem parabo- 
la eft quam defignaret fi cum impetu horizootali opponunaex 
vertice ipfius proicifhim hifkt • 

Manenteeadem conftru(^one> &£gura, dico etismpofl 
culmen» (lue verticem/i mobileex ^ proie^in, in eadem pa 
rabola continuare motum fuiurv« 

Sumantur 6^ p^ApCiim gquales; 
erit defcenfus / r . quinq; temporum vt 
ij. & qt. fex temporum vt 35. Sed 
cum Bf. (it i5,ipfa pf. eft 24» & qm 
32. Reliqu5ergo/r,/ir/,funtvtumim-. 
&4.&C. Quarepun^yjr,/. perquig 
incedit mobile funt in eadem continua- 
ta parabola in qua funt e , Scfi 

Linea etiam curua » que defcribitur 
a mobili fecundum quamlioet dire^io-» 
nem deorfum proiedo , parabola eft9& 
eadem prorfus quam deiKxiberet mobi- 
le fi horizontaliter concicatum a uerti^ 
Ce ipfius proiceretur « 



Manente eadem figura propofitionis 
tcrtif • Sit linea proiedionis deorfum 
fa^ 4f. » & fit impetus idem qui fiierax 
in proie^ione 4L furfum « Manifeftum 
eft quod mobile fi motu ^quabili moue 
retur percurreret lineam redaro 4f. Su- 
manturiam dhyhf. ^uales tum inter 
ity tumetiamipfi ^/.patet etiamqiiod 
ipC^ 4h.hf. motu ^uabili, temporibus 
fqualibus abfoluerentur cum ^uales 

iim t Sed quia grauitas fiatim incipit de- 

• ' orfum 





LiBer Secundus. tsp 

«Mrlumtrahere , mobile a lin^areda ^/deuiabit» & erit defcen- 
fus tg . unius temporis utunumji eritqs f qualis ipfi /r qui fue- 
ratdefcenfus uniuse^>ru(ndemtempQrum. Defcenfus autem 
j^J • duorum temporum eritnt 4.& fic fbmper deinceps . Qijia 
uero ^ 4 « eft 1 60 erit ir ^ « 24» & addita ^g , tota eg erit 2 5. 
Eodemmodoj/jfeft 32,&/^^4.er^otota id. eft j<5. 

Cum itaq. iint ^quaies ih^h^ei.ococ.hayeg^id^nt^s^o^ 
tinuato ordine numerorum quadratorum,)ut 9. i ^. 2 5. 3 5. erit 
linea c dgd.nonio eiufdem continuataeparabol? : ergo linea 
curua qu^ defcribitur amobilideorfum proiedto parabola eft ^ 
& prorlus eadem quam defcripfiflfet fi. a uertice ipfius, cum ho* 
rizantaliimpetuoportuno proie<^m fuilTc t • 

Diximus c$im h$n^Maii impetuopponuna^ quiafimobilc 
cum eodcm impemfroiceretur ex p horizj>ntalit€r deorfum.^ (^ 
ex zfecundum ^Xfttrfum^neifudquumeandemptarahoUm de^ 
fcriberet Vitraqi latione ^Requiritttr enim maior imfetus infro^ 
ie&Mue tx ^ftcrfumfaSia adhoc vf eandemforabolam defcri^ 
batquamdefign4uiffetfiex p hortMntaliterfroieCiumfuifiet. 
JELatit vero vnius itnfettts ^datium vt eademfatab.euadat^^erit 
bac ^ 

&imobile horixjtntaUterfroieCfum ex p. quolibet imfetude^ 
fcriffit faraboiam p c a» adhoc vt ex tifroieiium defcribat ean 
dem^dtbetimfttusex a adimfetumex ^.effevt an ^^ am. 

Tttnc enimfimobilt iuxta tangentem a nfroiciatur cum im^ 
fetudiifoeandemfarabolam acp.fercurrct. 

Si quis autemyfrofternmmerorum dfflicatimem^ ettquuati^' 
tulimus non demonUrutiwtmftttet yfedcomfutum , vetexem^ 
flum^ habeat hic demtnfiratianem furam^ framiffo hoc lettt^ 
mate. 

Lemma^. , 

Sijucritvtt a c ^ a h.fotentid:, itac e adh cf .. 
longitudine , drfintfaraUtla c e , b d » DicOi ( con» 
iunSik a t) iffamh i.medidmfrofortitnaiem eff^ 
interduas ccyhd*. ^ j.. 

JEfienim c e adhdJiim^mdHte^tz^tk^h^ ^ 

. V V j s bo^ 





1^0 DembfH Fmt^srum 

Siverh^vtfunt zh^zCt2Ld.pote$§-^ t^l> 

tii ) itafuerint pdrdUeU b e > c f , ^ g. 

UHgitmdine yfintqi a h, i g • stqudles^Di 

eoetidm^htyii^aqndUsejfe.Efienim J^ j^ — "^ — ^ 

ga dd^i v/ gd ddic^ vel per prdce^ 

dens yVt ic dd cf. ergo dinidendo erit 

vt %i ddia^it^ iiadic.eodemmodooJlendemMseJfe vt gh 

adhz^itdhtad ^b. HisdemonBratis. Efi £c ddhe.lom^ 

gitndineyVt ca 4i/ab,Wia ad ^hiVel %h dd h^velbe, 

dd tb.fotentii. Ergo he mediaefiinteric.ht.Iterum. 

EH ic adhc longitudinevt ca adyi b , vel i a ^^ a h , vel 
a i ad igjvel cf ad fi fotentid . etgo ii mediaeflinter cf , 

be. Intere4^demver)fmediaeratetiam tb^ergo fi^ eh. 4« 

quaUsfunt. ^od erat prdmittendum\ 

Refumfta iamfigurafnropo/itionis /. >&isriir/ yfiAt vt anted ex^ 

pofitum eBproieSfioper lineam a c d e f • ducaturq; reC^a a f. ^ 

dccipidntur aquaUs ap.fq. Eruntokdefcenfionem ttatttraU* 

ter acceleratamfpatia 1 c. m e , bf • in dupUcata rdtioue tempo* 

fum aU ^VA^^ih. drideoperpracedensUmmaaquaUserumt 

C p , e q ydrpropterea remanent aquales I c , n e (nam tota 1 p » 

toti n q • aquaUs efiyproduHa enim i d r • erunt aquales a r, r f; 

^ ipfa i r eandem rationem hahehit 4^ i p, c^ W n q , /yrM^« 

qudmhdhet ra ^ ap.velrfdd fq .) Ergoomnestinea qua 

€x fverfus i dfuccejftue defcenduntaUned f h .funtaqudles 

^nmibusefrfinguUs iUis refpeCHuh^ quaex a 3^ 

^/^iir/ i^/TM^ i d . fucceffiue defcendunt kli^ / 

nedzh^finguUfinguUs(quodenimpSenfum ^/ 

tftdefoU vte.ofiendipoteSideomnibus .)Sed k o i^ 

#mrr/ /5 ^ , qudrumftries ex a incipit per fup "^ ' 

fofitionefunt interfe Ungitudine vtfunt a 1 » 

a.ni , a b .potentii , <rj;# ^/i^w #«Mr</ //Ce ^jri* 

fum erdo incipit ex f. erura hngitudine vtom ^ 

-^^ itkffo^ th .potentik . ^dreUnedenmd 

acef. 





^ L$^ Sicimdiif . ^ fst 

ti$$femfr0ii&eieddefkfdr^ebi efi^ ^m^tmdefign^etfi ex vmi 
€€ f cMm imftttt B$rit^nt4U efertttmfreieSfMm/kijfet . 

Si mebile freielftim , dtm$ forM^eUm a b c fer^ 
ettrrit^indliqmiffiitt fUM&e b. emni grdmitdie 
/ftliaretnr^ tunifriiuldtibieferUnedm reifdm b 
d^tangentemfdratiln Utionemfitstm cetinttdret 
motu femfer difuulHU . SM/mdequidem demftd ei 
effettmnisctu^dquMfmtiim MtinfieBere fi/fet^dmt dccdera* 
rcy velretdrddre . Mdn^fium etidm iB^tmfitum^fittxmibi^ 
^ lis in qudlitit f^niini tdngintis bd> iundimfemfirfittttrum 
firequifiiirdttnfutt^ih. 

PROPOSITIOIF^ 

IMpecus iti pun^ parabola^ vt foiK p<utione$ tangentiCi > ii^ 
ter duas parailelas diatnetro intercepus • 
Propofita parabola dkc* ducantur e, i3> 
tangentes di^igych.^ quibufcunque 
pundi^ d.h.c. tum daa? linea? par allel^ 
diametro vbicunque Gxiadiyib. Dico G 
lineas interceptas di ./g^ i h . iplos im . 
petusqhifonrinpun^ ^tf^^.^.propor 
cione reprefentare • 

r Vnaquaeq. enim ipfarum d.i ^fg, i h .eodem tempore abfol* 
ittretur a mobili)qaandoquidrpr ogreffio borizontalis qine tn- 
ta* duas parallelas ^ th.e&s eodem fentper tempof e debeac 
abfoloi» vbicuaq« repertttur loobile , & p^qttamcunq; incli^ 
nationemprocedat. Sed motusinipfis luieis interceptisfuiat UmfiJf 
«quabiies «ergoimpetuseruntvtrpatia* Quare impenis ipfius ^ 
di, vel punifti d. erit vt linea di . Ipfius aittem;^» vel pun^ 
f#)etftvt/^. dclicdeiaoips. QuoderatSoc» * 

X PRO. 




{^^ Demm7^me0A&m 



«Vi 



,PAP P O S ITi.Q f^' 



IMpttusinpundlis parabola? a^qualit^r vtrimq; a vcrticc di- 
ftantibus , aquales funt, inctr ic» licct alter aiccndat , aitcr 
vcrodefcendat^. .; ,. 

. Sumatur in p^abojla proie^onis fa<£te vcrfiis. 
^ &r,pundaqu«uis^^r»quas\aequalit^ diftcQt 
a vertice t » hoc cft j qus (mtin cadem horikOQ-^ 
taii linca 4 c • Dico impctus in ii & r aequalcs^ fif- 
U • ;Accipiatpr4/a;qMali>ip&^4/j& ducaniEur 4 -. 
e»f4iiqu«^Hmvtraq»i{Cang«j)scrit^Du(;aatiirctia , 

W« >fj(%g^^ diamctrop^rajlelfi^unqstiilum ^ ^ 
foerit i &' producantur tangentes Ah^4g^ Erit cr . 
go per prseccdcntcm impetus in d vxfh. &i0V ut r^, quasfi 
^quales fucrim^aequalcscruotimpetnsjnpuniftis 4 & r. 

Latera ^^,i/r .funta^qualia^&i/^commttnc; anguli aute 
a^i^rc^i^icrgpanguli ^if^f.^/^i^funtfquaks» Anguloaiir 
tcm ded ^qualisi^u /A^.obparallclas^&ipfi ^f^. ^qualit 
t^cghy itcm ob parallciasscft^ecgo triahgulum r ij'^ ^<^qii 
rc^linca/f.bafi parallela, ^arcyijf,*^ ^quaks fqnt* . 
Qupdcrat&c» y . ♦ 




f - 



Coroll»riarti> *- 

Hinc c0lligeTef»fMm^fMi^4fm<&hntMh ZyqMedfrmfhi^ 
leexfsiftilo crefleilatMrretrcrfumfertdmlfmffuimvidmcmth 
iai^iktedewh d^dtmqi MreJ^ituee qMmh^et iMfun&o c yfit 

:4wSi^f^m ip^fHmh&AAndill^iUf^ianeMfit^m bAehM 
iAX^^K^^^f^^^ftAm defiffymi 



1^4 



' t 



s 



fteiuieiKlurn«ftalit^aciopninolibro,eimdetn efTeini-; 
-C . ' .' petum 




petiii»paraboI»inf'A^acj;nuiscadcatisnati]raiicer6K pQQ^ao 
iubiimitatis i/yufi). in '<«.:. 

Ducanttir tangentes 4 ty h/,^Gtke 
parallela ipfi d.c . Nptum ei^ «ft per 
4. huius itnpetum parabolf in ^. ad*im 
pemm parabol? in << , dTe uc />/ ad ^ ^, 
cum iint &/,8c4e inter eafdem ad dia- 
metrumparaUdasiQtercept^. Agatur 
per 4/ & « aiia parabola </«/^ . &impe 
ote C4&^per '<(/<«, adin^mmcarus per 
jf^»«rifttt^Jpplicata ^« ad applicatam ci(. Si crgo fiierint 
^qualestangens V^&appplicata ^*,erit€x^uo,iinpenis in 
6 ad impemm in ^ ut ^/ad cL nempe aqualis . Oftendo 6/ 
yl^, ?qualfts tflc, fic. Secetar ci biferiam in /, & erit per 
demonftratainpr§cedentilibro,f/ media proportionalis in- 
ttt Cii>6c dd^ ilamfic.C^admcum caa^ re&bktmAixmcdd 
fub eadcm altitudine', eftutriiadjf^, ergoqnadracuijl^j^ 
ad^uadraaim ci eft uc r^ ad dd. Siimpttsautem quadrjito- 
rumquadrupliseritquadratum/r adquadratum ch\Kcd2ii 
dd.^ componendo quadrata/r, c ^, uel quadratum/i^ ad qua 
dramm ct eritut cdzjidd^ hoceftucquadracum r h ad qua- 
dramm de: icdi quadrata «r^ &^r. fqualia funt , ergo euam 
quadrata/^, ck QuareaEqualesfuntlihe^/i^, r*. (M>d 
cra't&c. 

PROPOSITlOril» 

SI ab eodem ptrhao , cum eodem imMai , & eadem dire- 
dfone fiantproiediones utrinque t lUMftim nempe , & de* 
orfumrmobileutrinqueperpordoncs untUs eiufdemq. conti-^ 
nuatap parabol 5 pcrcurret. 



FiatexpunAoi#"Cimidii«dlione i#i.pft>fe^'ofurfum dc. 
drabeodeiiipunaodire^on^i^^ibtproieaiodeorfum dd. 

X 2 Dico 




DioofM^ymm, !Sc duKJmcoKOTiafanTfwU 
rabolain efle •' Si enim condnua noii.^»deinic« ^ 
tatur mobile exvertice ^verfus ^.perparabo 
lam cd. Tuncniobiledum^iiV/.non fitcon- 
tinua parabola» non per ipfam ^f ^> fed per aliS 
lineammeabit»qu2fitifr« Vcrum mobile iiu 
pui^iStb 4 eundem habet unpetum fiue ante afcenfum s< # fiue 
- poft defcenfum c d . Mobile ergo ex pun^ 4 . quando venit 
/^ eic r meat per r^i^, quando vero pCDiciturex d cum eodem 
impetu 5 & dire^one curritper di. Qiipd eft abfurdum* 
Cum enim in vtroque cafu difcedat ab d cum eodem impetu » 
cademq;diredione> debetetiaminvtroqscalupereandci^li^ 
neam 4^ ambuiarc« Qupd &c» 



P R O P O S 1 T I o r l I J. 

DAta qualibet parabola d mobili furfum prokfio defcnp^ 
ta > proie^o perpendicularis furfum eiufdem mobili$ 
i^Qcx cum eodcm impeni » tandim akendet , qiian:am cft aggre 
gatum altitudinis t & fiiblimttatis fimul datx parabol^ . 




V Sit parabola dh^ cuius altitudo c t^ &c fubli- v 

mitas ^i/.ponaturq; 4r aBqualis&paralielaip* 

^cd. Fadaautemfitparaboladproie^ionc 

, ex 4 vcrfus i . Dicofifiatproiedio cumeodg i 

impem perlineam d< furfum,mobile vfq;ad 

piindum € peruenturani elfe • Impetusenimparabol^initfyfi- 

ir. rf4^ uefiatproiedioex ^»in ^,fiue ex ^ in 4, idemeft^vtoften- 

hS. diniftis . j Ai ex Galileo tdem eft ac naturaiiter cadentis ex J ia 

f y i^ii impetus naturaliter cadends ex ^ in r ille eft qui reudiir 

mobile ex ^ in d^ttgb etiamex 4 in r • Qijpd&c. 



J^fiifpJlnumq0Umioddii9mim^€t$mnw^^ iiiumm 
/fdt^sd€t€mms^$mm^v9(Um$wf^ idUdmimrdtim 



\ 



II 




rArf 





dMM ab. tMmftufusmfiififi. Sitiimfttisdsum 
tmtttts ftumtms ri^surittttkd frmiemlttmm^Uk cx t 
yffituifstmmMmftmiStmferftmiicmli h. Simt^ ftm 
i(m€fi^Mimtmtfi^tti^tmttmm^ir(t4td€ttHS€tt\i 

Leinmayi 
Circ4dim$€tftmt zh. fermerticeM a> & 
qmodmis pmmCfmm c • mlid^ mtq; dlidfdrmboUmtfm 
cmSitmctmr. Si^mimpff^ilccfijfimtcircddut 
mctrmm a b fcrfmm04 a c^ c .dstdfjstJtholtt, & 
cx c ducatmrtrdimmtim c h.tmm dlid quttlibct 
mrdimctmr df. S^udrdtmm crgo cb, dd dmo 

illd fmddrdtdimdfffdlidfdsfecdmdcmratiomcm/jdict, mcmft 
^mambahct bsL,dd ^L ^od cfi dbfurdum. Ergo circd di^ 
rmctrum^su , .. ^ 

f R O P t^ S I T J O t X. 

DAtoimptm bd rhQceftquantw eft naturaKr^r cadentis 
ex ^ in 4f iuxtd definiuonem^ dataq. direftione a /, iux 
ta quam faciea d a proiet^o cum impctu dato . Oportet ampli-'' 

tudQnem, altitudiacm,totaa>q;futuramparabolamhuius pro- 
ie^onis repcrire . 

Ducantijrpcr iT &; ^ horiromalesli- 
nc? dJ.iLQc^u fecnicircuUis dfh cir 
cadiametrum 4^,quilineam4romni« 
no fecxibit > cum ipfa d ^tangens lit . Se 
cet in/, & ducatur fe horisontalis , & 
producatur/^ ?qualis ipfi /< . demura 
agaturper jrperpendiculum Igd. §m iam circa diametrum 
/i/perpuniaa^ & ^•parabola-r^,qu«vnicaerit perlero. 
maprgcedens.neqj.alttpwabola^rcadiametru^ ^ perpun- 
W S^ * d^Wijfomtk Dicohanc fiffcpatabolaai qu«fi-- 

« tam» 




tangat parab^lanSvi v.<*. £ft «oim r^Hid W.\ipfi«sy^^^»IWr 
plapercc«»flni^«l^in>&id«(y^iiale*fanx ^t^S^^ ^u^*>i 
tangenft eilv. > .- - »i v: .:k .a;v^.v -i. .^^ '-^ *- 

InfupttA PicahaoojiwttAoi^m ab Miq!*jtf<J*o<*«^**J'^' 
Sunt enin|.4t t ^^Ak,&i»m^^i ^^^ytg iftlcitt^i f/ 
femibafis, & ^ /, in continua proportione .♦ quare ^/'fubUqai- 
tas cft ('per 5 . propof. ^ euu Cor*llarium Galilei . ) 

lam (ic . Impstijsparaiwl? -45« iu puna» '-4 untxts eft qua 
tus naturaliter cadeniis cx / in </. per i o. Galilei . hoc e ff^ex 
Mn i<. fiuepcoic(aiafcend<^teisvrx rf in^. Habetergopa- 
rabola inpuncao 4 ctianMiii^tMn dat«n . Quare fe<ftum cft 
quod&c. . . • . 



> f-«.V 



StdquU h*e prtpofim nugni erit monifnti profeifMOtfH *S 



i^^^^ oBen^dmus etidm alio modQ , 



i M M 



\n 



.4-»''* 

* w 




Kl'^ 



Sit impetus datus idem ab.?>c eadem 

direfitio afc Qiweritur parabola <5u§ * 

fietabhacproiedione. Fiat ut ante cir 

ca diametrum a h ietnicirculus» qui it-- 

cabit lineam a c , cum dd^ fit tangens . 

Secetin /idudlaq.horizontali efgixsL 

ut^qualesfint r/i/^,defcribatur» fiue 

tamquam defcripta concipiatur parabo- 

laperpunda Ayg^ circadiametnim^^^^ DicohancefTepa- 

rabolamproieiSki, fi a pundlo a iaciafttr,iuxta dire^ibnem d 

^, cumimpetUii^. Nifienimcumtinobileperhanciamdt* 

dam parabolam» curret omnino per aliam > qus fit 4/ . Repe« 

riatur uenex» fiuepundhun altius cfteris huius parabol^ ir ^, & 

illudfit/« 

Patetprim6qu6dpuni5him/e(renonpoteftin linea Idy quia 
cumiineaiirtangatutrainq.parabblam5fecaretur id axiscd 
munis bifariaminduobus pudditi uerticibu^ parabolartttny 
abfurdum.NequepoteftejQfeinlii^ea^^* Quia^a^ peiriier* 

- ticem 



\ 



I 



abfurdumrfoUj^qim /V^fi^caairixi&riapicx omnibus (ibipa* 
ralldis laaQgul^ f 4^» . 

Sh xampjmidiftB / . vbicqnq; ducaturq^ / r/. horizontalis . 
Qma fn^fmj. fiint aequales^per fecundam nuiu^er unt nr^r^u 
& /^r f^/fv. aifqpl(?f i & qfiia jpa^^^bola >^^: imperum haber 1 4 . 
bpc ^ik^m^isji^A puniftitfniubiublimitatJs » &ideo lineas oft 
pr^fm.m continuapropojrtione erunt ; & redangulum of m . 
quadrato / r sequale > coiiimutatifqs lineis cum iibi ^ualibus » 
, re<ft.^gulum/iy^a?quale erit quadratp /> .Puni^mergo r eft 
in femicirculi peripheria • Qupd eft Abfurdiitn,rei%i enim linea 
4/indiK>bu9tafmu»>pitf^speriph«^^ Quare&c» 

' . '. • • - . . - .. . ' •» 

„ . - /CoroUaria.^ 

. ^if$cmd»^!(4i^ifmji^jd4tCimfitit4UciiiMJmd 

cMiU^JiKvefki^f^id^^^ Sidefcrikdi0rcir^ 

ca e a ftmicirculus a d e » d^^i dltiatdiMs^ & dm 

fliJtftdimm^ ^^^ d^eddem 

tefemfer eodtm^jmftiic^ ^sfdHtf^oieCiionesfer 
lincOs diKtjifiiiiodfi, tleudtdf^ ^ c , a d , a b . Proie^ 
cUq^iid{eCiih4iM^4i^^c^fi9i ac j^endetvfq^ ddborixsm^ 
tdlem fcfrodffbHMi i idihis dst$emfd&Ms iuxtd dire^ionem a d 
dfioemhaie^it ,M Hn^M , b.d^. frodnSd • Froieiiionis dntemfe^ 
cnndnm lincdm ii\>/d&d^m^imd4ltitnda drit sn horizdntdli 

gb frodnU^A^ . . \ ^ 

In liho (^fipde mi^stpMmdUitrMceltrdto ofteditnr^ fro» 
ieffddteodfjf^i^if^fifn^ lO^y g^JfJfidtfis dinerfimode inclindfis 
fnlcidntnr/emferddvnumidemqyfldnnm horizdMdle f^t-^x 
nite.HicvirodffjtretfingnUsfroUSlornmMfcenfifines vdtid'^ 
ri^ ^andoferdmetnfnr^mfinf mihfnb'ti£tofnkrofroicinntnr 
inxtd dinerfds cieuAtiones : Minns enim dfcendit mobile qnod 
\Ifni/^^zh^ nHn0siJ^n^4^'imittitiQr, fnnm ilktdqtuldfer 
M9, ^ jC^ «iff ^f << cUfUfUikfiriieihm^ ""..}. 

j. f !^ W9^ jWSte^fe^»^» jtthodfedndndfofc^ vtjd : 




Um6nisf€tpndic^i^fkmn^ 

Manifejium etiam efi ^amplitudines ommifem^ Amgeri^ 

i£^sct^erhfai£am adartgutkhtA^.eifttm; - . **- 
.- ^afitmreaa-^tYo^vfjueadf^fef^ 
dvnec fertitur^tttikttefca^t \ qtto^-dccidiimfrUii^Mtferftttdi 
culari, qua^tiMlamkabciamfUiVtdirteUl' ^ '• '^ 
. Dettique^bftYuatre ticet amflitudirtesforahlarum db nekm 
imfetufaiiarum^ijuarttm eUuathnes aqualitefui attgulo ftmi^ 
rehitdifiettt^itttajeaqualeceffei^ \ - 

Cum ertim Ituea a b , a c * aqualiterdPHetii 4h eleuatiette fe^ 
mireCfa^ erunt arcms d b , d c aquales . quibus infifiuut squoles 
duguli; (jr ideo arcus b a , -C ereiiqui ex quadrautibus fquales 
eruttt , ergo etiam^fitms eorum b g , C f^qthdts eruttty ^ftofte^ 
reaasnfhtudinesintegr^farabolarttm ^quaquidem quadft^U 
fuittfinuumh^scLerutotaqudes* - -' ' ' * 

?atet etiamfrmBionmtt aqtidiief' kf^ft^reBd difftttttimto 
altitudintSyf^fubiimitaiesreciproic^ittterfe ttfitdUs ejfe ^ kn 
efi altitudinetn *vnius , fubiimitdti dlterius aquari . 

Corollarium ergo erit quod GaUieo Titioremafitfis arduttm 

fiteraty nerttfefroie&ionemfemireSiam omium maxtmam effeak 

eadem imfetufaClarum . Si enimfortatur an^ulus c 

a d fetnireSitte erit c d femidiameter , hoc efi maxi^ 

ntus otttnium finuum qui in/emicireulo darifojfint . 

Patet e^tiam iutegram amfiitudittemforaholafemi 
reCia duflam efie iinedfitbiimitatiSjfiue imfetus a b 
qstia demenBrtttaeU quadrttfla re&a cA^hoe efkjltt^ 
flaiffius;ib. 




pRop^oirrio x. 



»«^ .• 



DAto impetu & ^titudine imietuenda fitdfredib ftoAa qui 
faAa fuit proiedio: inueni enda euam fitamplitUido^. 
ie^oois . Sicin pnKedeoti figura , impecils 6ms^^ «^oata 

altinf- 



akitudofiCiir.fiatcirca 4^ {emicirculus ^i/^,&clucantur €dg 
horizontalis y 4 d autengi ad pundum d. M ani^^ftum eft ex pr$ 
cedentibi|s,dire(5tiGnemquae(itam eiTe 4 ^,amplitudinem vero 
integram eflfe c d * quater fumptam . KuUa enim parabola prf- 
ter illam quas fit iuxta diredionem 4 d. cum bab tat impetum d 
i > habebit altitudlnem ^c. 

\.\,.PR0P0SIT10 XI. 

DAtoimpetUy&amplitudine inutnienda ikdire^o iuxta 
quam fada fuit par abola ; inuenienda etiam fit altitudo • 
Sitdatus impems 4iy8cCit4d quarta 

pars data? amplitudinis • Fiat circa 4^ 

lemicirculus 4ci,8c erigatur dc e (quas 

fiin femicirculum nonincidit problema 

impoffibile cft J fecetq; femicircuium t 

in pundis^ c.8cc. Dico vtramq; dire- 

^onem fiue ^^ ^>fiue ^ir, fidatusimpe- 
tus it^,adhibeatur,paraboi^diefigna- 
re^cuius amplitudo quadrupla erit lineas 4 d. Hoc enim ex prf^ 
cedentibus liquet. Nam proiediones fa&x cum impem 4^ 
iuxta dire<5tiones 4c vci 4 e amplitudinem habent quadrup» 
lam ipfius ^ < > vel/Sr . veL 4 d . quac inter fe ax]uales funt • Al« 
titudo vero efle poteft tum linea 4/y tum etiam 4g . Vt appa« 
rct&c. 




PJiOPOSITJO XII. 

DAtaamplimdine ^^» &diredione 4^«inueniendusft 
impetus , & aldmdo parabol^ . 
Datisi)fdem>inueniendafitmenfura linef perpendicularb 
ad cuius apicem afcenderet mobile fi cun eodem impem fur-| 
fiim perpendiculariter proiceretur # 

S\mmi4d^ quartaparsipfius4^9&erigaDturpeipeiuU< 




17 o D^mtu Proieihtum 

culares dc.de. fiatqs ahgulus 4c crt^ ^ 

ikus. Dico itf^. efTe iitipetum proic- i 

dioniSj Sc jdc. altitudinetn . Semicir 

c|iius £nim circa diametrum d e . tranfit i 

perapguium redum 4cc Ergo para- a 

boise> cuius amplitudo fit a ^,& dir e^p 

dc^^m^misdk^d. 
Patet etiam altittidlnem para bol? *ue lineara V r . vel ^ / • 
Cumautem impetusfit 4e manifeilum eft proiedboneoi 

perpendicularemfurfumex m faiaam afcenfurarfi efle vfquc 

ad € jiuniftumjiiproiciaturmobilecumeodemimpetuaq^ 

fada fuit parabola jit. 

£x h^c^prepofititnecoUigerefoJfumns qu^ntU ffdtij ^fceto^ 

j4tferrensgUtns ^fiqn^ndo 4h dneo .qnaUb^tormento fnrfnm 

ferfendicm4rUeri4ci4tnr.:cninsqnidemff4t^ menfnrn tnntd 

erit y aftexnnUdferfendicnldrinltitMdinefineArte^fine ndtnrd 

fdSid^^efrehendifoJlftt \4nt nliterexferimentofnhidcete . 



\ 



P K P ^ S J T J O XtlX. 




Ata altitudine ^ ^ , & dirciSbione A c . reliqua^ 

reperire . 
-! i Ducatur perptindum i. horizontalis hc xju^ in* 
n^^inipfkm 4#inpundo^^. Piatq;.an^us dcd 
xe£k\xs: & circa trian^ulumredan^uium jicd* tran B^ 
iibitfemicircuIuspropofitiQnisp.huius. Ampiitu- X 
tudo ergo quodri^Ia erit ipfius h c dcimpetus icat 
jid. Qupd&c 




T R F O :sjr J O XIK 

• * ■ ' 

^ Ata alritiKime if^, & .bafi (cuiusmicn qiMT- 

taparsfit/(f)^diquaieperire.. 3 

Coinpleaturredbuigulum.A4 c.d, 4&.<liaineter .d j^ 
^ !dircctionemipdk;abaJFactoxlein(le.angttlo4</^ 





:recto 



redo , erit dt impetus , vt ^ile ex praKcdemibus colligi- 
tur&c. , , 

FkOFOSITIOXF, 

PRoiei^operpendicuIaris/urrumaBqiialiseft dimidiae bafi 
proie^nis (emke&x/G fiierir ab eodetn impetu fa^a« 
vtraq; tam perpendicdaris , qu^m femire^ proie^o . 

Sit parabola f emire^a dbc^Sc fuper 
dd media amplitudine fiat quadratum T 
d def^tni d e dia^eter ipfa direx^p fe- c 
mireda . Fadoq; c jrcay^/Iemicirculo» ^ |/ 
tranfibit femidrculus per i centrum qua 
drati , & erit df impetus j quare proie^ 
dio perpendiculari^ iiirfum vfq; ad/pun^um af cendet 
tetegopropofitum* ' 

F R p o s I T I o xri, 

SI &^fueritproiedioadeleuationcfflatiguHremit 
amplitudo integraeproie^onis erit latus re^um def 
tseparabolx,^ 




• 1. >* 



Sit eleuatio lemireaa luxta lineam dbt hJBeim. " 
que fit a profeao qu^Iibet parahola dcd, Dico yp 

<r</.e(feiamsre^mhuiusparaboI«..CjumciMm ^ P^ 
angulus e d i, (emireaus fit , dtb reaus., enint X 4 j 
fq«^ia4atera 4^, «^ . ergo^e, duplaeritipfiiis 
«r.Sedcum ^emcdia^MropoitippaUs fitittser latus reaum ^ 
* tf. eritlatusreaandupliuBipfius ^ iMMpcauuale ipfi dt 
Qipderat&c» . , : , • . 



/ 



PRO. 



17* 



De mom^^^irkm 



')3i 



A 

beat* 



P £ P O S I T I X y I I» 

D proiedtonesjequalesfaciendas, tninotimpettis 
quirtcurin ea, quaeaddeuadooemfemiredaijifierider. 



^'- i 







Demoaftratumiam eft , fi ab eodem impe- 
Xa h&x fint proie<5tiones , iongius procedere 
eamquaeadangulumfemiredhimfueric explo ; 
fa. Sintproif(^'ones<<^.{emireda,& e.e, 
non femireda. Dico impetam ipfius r. c . nbn * 
femireia» maioremfiiifle quamipfius ^. femirea«. Si enim 
fiiiiretaeqnalis,tuncampiitudeia(9us df ex demonftratis fiiif 
fet minor quam 4e.vt yerbi gratia -«/j fed<nim aequaKspona- 
tur&mplitudo,maioromninoimpetusfuitper ^r. quamper d 
hi veiminorimpetusrequiritur infemire«aa quaminalia» 
Qgoderar&c. 



^» i' 




• * 




PROPOSITJO XFIII. 

Empus , fiue durationem vniufcuiufque proie.5lionis de* 
iinire. -^ f-. .. : 

. 'ConflruAd folita praEcedentium fi- « ' 

gora,fitproicaio4^f.oponettempus, ' 
fiue durationem dus reperire; hoc efl E 
.quantoteporefiat ktio ^r paiabolam 
dic, Scimus iam ex Galileo idemtem 
I>us«flelaiipnis4(^«-., &<XKl6ndisex-^ • - V 
in 4? . bis . Poaomus crgOit«npus<:adieittiS ntituraliter cx/id 
4* elfe/ir i erittpiOempiKp^ led niedia propordonalis 4g, St 
h£fc ^pijr. ecQQJin&mg metitur^mpus-facionis per r^, fi- 
ueper)^^,^perfem^}araboIam^r,verperimegram eria 
|>arabolam 4^^, Eandem enim radonem inter fe l^bebunt 
durationes parabolarum,quain hd!>ent foniparabolaifi t & oos 
logwaiur <leprqpottiQDtf)us« QoadeiiieDfijris* 

^ y PRO- 



1 



L$$er Sicmabii . 



>7J 



f R O P O S J T 1 O X/X, 

» 

DVradone$proiedi<miim{iimvtline$ordinarim applica-^ 
t% in aliqua parabola ad fuam vniufcuiuiq; aldcudiaeai # 

Sintalcitudincsduariimparabolarum dt^ 
Mc.(z quocunq;impetti fiue eod^m, fiue nd> 
factasfint» &quaicuiiq; bafes habeant» fiue 
^qualesyfiueinfquales.) Fiatcirca r^i.pa- 
raoola inuerfa d/d^ & f emicirculus . 4gi. 
Eruntque tempora parabolarum vt funttera- 
poracadeotiumper eji^ ^^jhoceftvt cd^ 
//'per X pr^edends4ibri . Quod erat&c. 

Suncinfuper ^^,^itf. ineademradoneac cd^hf, quare 
etiam^^ ^j^ir.chordaeialemicirculo erunt vt tempora para- 
bolaciimd£« 




T Ro r o^s I r I o x x. 



PArabolarumas^pulembafim habendum impetus in puncto 
fublimiori funcin contraria radone temporum, fine dura- 
donum earumdem » i 

Sint dux parabol; dc.nf. qnaseandem 
habeantbafim» &eundem axem bc • fiat<^ 
circa^^. parabolainuerfa^V^. Dicoim* 
petumin c ad impetumin/*effe vifdy ad 
CM. Impe(iiseniminpuncnsy;&/'funtpu<^ 
riilIiim(>etlisiionzontales»ibcuiidamquos ^ 
conficituiiacioiiori^cont^ -4^. 

Cumltaq;eademiadohori2ontalis itf^abfoluacurtemp6rtbtis t^t^* 
rr</^>emmin^etushorizonMles redproci^ut/y ad rr^^ 
per3«Propo£(^DeMotanawcaliterAcceIeratD« "" 




\. ^ 






,-,*».' 



. »* 



,• ^ , * . 



pitou 



t 4 



.« • 



^74 



Dtimim^f^9&mtm 




p M p a s i T I o j X I4 

IMpeturflpurum horizontalem, qui inuariabilis eft femper' 
idemiii urioquoq; parabd^punccacie&iire* 

It em etiam & perpendiculacetn uariafasiiem . Re 
petita prsecedentiumpropofitioQum H^iica % (jappt^ 
ijimus , ut femper , impetum totalem^ fiue compofi« 
tum pf oiectionis , quem habet mobile in oiincGO 4 , 
eflfe tamquani naturalirer cadentis ^t.iin,^* &- 
hunc ponimus elTe yt iinea id. . . 

Siti^i^altitudoj&^i^fublkmtasparabol^ Er 
g6 iinpetus cadentis per id fublimitatQm paca|>olg,»'emutli^ 
fiea^r,mediaproportionaIi$incer>i^,^u/. ... 

Atifteimfwjtuscadentis cxk m d eftiflcfurus horistoiita- 
lisqui latipni ineft iaquolibetpuiicto|)ararbolij , & cftimiarii^ 
bilis . Quare in vnoquoq; puact6 parabol^ impetus horizonta^. 
liseritutlinea bc^ 

PerpendicuIari^aeroinipeQi^qaiai^iBp» v 
molationispunctofic determinabitur* Ma- 
iierite femper unica fuppofitionerhiipetum Ql^ 
iif et cafus per < ^^ ^ elfe ipfaro . e d. Impetur 
perpendicularisinfineparabolae h^ eft tam»^ 
quamnaturalitercafleiltisex h in^^ uelexiir 

iri^. EftcrgoutmediapropomonaUs4^^ 
Qypd&c^ 

Sed cBenddmMs eiiamff» Cm/tdrk , qujmd» vMrieit» im^ 
fetusreffemiihAdhwix^tnemy. eittpiem gUhifenei 46 eeJem 

tmne»tefroieaiierefcifeHimimftutsfei^feMdieMUris.t$eMqitf 
ddmedMm crefcMMt eleMdtieMes twm^titiy MtdtitMdiMts f^rs* 

teU»fededrdti9MeqMdcrefcitiMfeMiieircMl4eit0rd4A{, HiMC 

MimddMerteteUcetfMfMrMmfMtevtidem gUbMsferreds tedem 
termeMte eitflofMsdMmadhmxAMtemredit dUqMsidoteSidfir' 

ttifefqjdetMerMmtrtticidttqMdMdeqiver^MeqMegldeiem dlicM* 
mldtMMdlfderefejjtt* 






■> 
/ 



UBer Secundm • 




V^L : 


K 




N 


/^V^ 


A 



etidm menfmt^ , JiHiiHdicjtsmom€M9YHm vemit4 
tis^ aggregdmumfer/pdtid b a , c Oy. 

* 

P R O P O S I TJ O XXII^ 

IMpetumcompofituin,fiueabfoiutum, quantus fitin quoli* 
bet pjindo pat^abolj demptnftrare y 
Infolita praecedentium propofitionu 
figura,fumanirquodlibctpuna»m.#m ^^1^^.' P 
parabola gAdM ducattir horizontalis 
^^/^•Duc4ttffq;r^.Dico(^ft;aa/em- jd 
pereadem/uppofitione impemmfilicee 
pcr tg eflfc tg ) quod hnpetus compofi & 
tus iri .#, fiuc in /, cft jreAa 1 1^ Gum 
ftiimimptnwiiipupaopacibolc >i ftytnafuralitcT.cadenws 
•cx pun<aoiiibliniitati5 r^vfqi in/, vei ex c vfq. in /• erit ilfe 
hnpctus vt r> medjaproportionalisinter ^r , tJ. Quod crstt 
&c. 

/^(fm etidmhocdlio modo eonfidcrdbimus . 
JSiexpnn&ofHUiini dlicmus ddtdpdtdholfMi^pdrdhldcir* 
cdednJhemdidmensmdefcriidtnrjlinef ordindtimduffd inde-' 
fcriptdydtttrmindbmitiimfetnsdbfolntos injingtdiffun&hf^ 
tdkotfddtf^ 

Sitddtdpdrdboldzb^Ci^nspgn&nm 
fublime fit c . Circdcommnn^ didme 
trumfdtper c pdrdboU qudMot c d , 
2>u&ifqufprdindtimquot.cnmjUineh 3t 
e ,fg. hi,bd« I>ioo imf0tttc input^ 
Sis z^f, h , b ytp^ofmthmf a e, I g- 
x!^h&niiMocjr^ot.^n4t0iic*u^itt^ a^f r M.paairfMt 

vtim* 





jf^4 7>e m^tu Proie^amm 

vi impetm c^dtmtmmper c a > c 1» c m , c n » ntmfe^^^f 



Lemtna. 

LtMed a b • fU4m im ff^cedemibusfref 
menfurMimpetusfonebdmms , ^ circA quam 
femicirculum defcribebdmusy qudridfjirs eff 
Uteris reSHfdrdboU b c ^fd&d db horizontd 
lifroieStione • Fdtet hoc exfrimo iibro^y cum 
zhft imfetus^ hoc efffublimitdsfdrdboU 
b c fecundum GdliUum • dttdmen demonHre 
turdUter. 

CdddtmobiU exzin\>y& inde horixdntdUt^t conuerfitm 
defcribdt fordboUm b c • Sumdtur b d • dufU iffius b a 
Ergo 'mobiU jemfore cdfus fercurret horizontdU ffdtim 
b 3 , eritq-y omnino foH temfus cdfsis in ferfendicuU d c. 
Sed efl etidmfemfir infdrdboU b c » ergo in c communi concmt 
fu erit . Cum ergofdiidfit defcfnfio d c temfore cdfusy erit d c 
dqudlis, ba • Pdtet dutem quddrdtum bd dcfudn reBdngid$ 
fubc6yetquddrufU ha(cumvtrdq; cd^b a/emijfisfitiffiusb 
d .J JBfi ergo b a quurtdfurs Uterisre^i fdrdbolf b^riMntd- 
Usbc. Sinod&c. 

PROPOSJTIO XX III. 

OMnes parabola? ab codem impetu d h 
£i<5t( idem babent latus redtunu (dum 
inodointeliigaturpun^m 4, ex quo fiunt 
proiediones efle vertex omniu obliquaru« 
parabolanim Sit horizontaiis parabola dd^ 
& non horizontalis df\ fumantur ia tangen 
tibus ipfaru sequales dc^de. 

Qjiia idem impetus eft per ^ ^, & per ^ r, 
ip& dc^de^ abftrada grauitatis operatio- 

QC I eodem tcmporeabioluerentur s eflentq. grauia eodem t& 




""V 



LihwSHtaMs. i^f 

p(»rein>.& #.'fedcttmgraurutscfpcrerur,&iclein(ittempus, 
cqtiales «imt dcfccnfu» efytd, Quadratum autem m c . »qua- 
tur reftangulo M>«dBc quadrupla d i , cum demonftratum fit 
in Lem. pr«ced. re^am s h efle quartaip partem lateris re^pa 
rabolf ^</,quareet}amquadratum ^«r aequaleeritrecaangulo 
lub f/;&quadrupta /ih. Eft igitur eadcm ^J^quartapars la- 
teris redi omniqm parabolarum ab eodem impem fai^arum. 
Qijpd crat &c. 

CoroUarium • 

Hinc mdmift^um efiftmfiYfuUimitdtemparabeldnm tfcen 
^tHtium yfiue liHedmimpetuSy qudrtdmpdrtem effe Uteris reffi 
UUus fmUuupdrdhld ehH^us , qud verticem hdbedt inpUH^ 
SofefdrdtienispreieSH di inMrumejut^ imfeiientt , 

Verbigrdtid . Stm^lefofi cdfkm a b ex ^uiett 
in a , conuertdturttem horizdntdlithr ^fedfer qudm^ 
libct imclindtdm b Cyfdrdboldmqi defcriidt bd.Pd 
tet tinedmimfetus yfiuefublimitdtem^h effequar 
tdmfdrtem tdteris re{fifdrdboidi hA.eonfSderdtd 
tdmenfdrdbold obliqud b d itdm eius vertexfh 
fun&um b , & dffUcdtdrumreguldfit tdngens h c • 

^od dutem hfc conuenidnt cum doHrind Co^ 

nicorumy fic demonBrdbimus . Sifdrdbold ab 

duds tdngentes hdbuerat ^c ferverticem y fi^ b c 

nonferverticcfn ifumftdq;fiierit^d qudrtdfdrs 

Idteris reCti , Hico inn&dm d c dMgttkste^hsfd* 

cerecumbc. 

Jgdtur b f ordindtim . Cttmfint dqtt/des f a , . 

a e« erit quddrdtum h f. qud^u^^Jum qukdrdti c afedidem qud 

dtdtum hiqnddrt^UtmefireajmguU fad, hoc efi re£fdnguU 

Qti^. dqudUdergofuntqu4drdtmmczy&teadnguUm$ eadj 
dngulufq; e c d . reBus . 

Hic,nififemitujdhsromofird.effi(yfdcimmieUceremus df^ 
monBrdtionemfoci . Si enimproductreturhd . effentferqudt^ 
tdmfrimi elementormm dqudles dnguU d e c , d b c • feddd rem 
nofirmm. 

Z HU 






li ^ 8 Dmhtu Proh^ti^m 

[ €usfit c .ftjfimstur sittUUbttftmQum a ittftifhtt^^ »& Mtii^ 
patittt du€4tttr a d . 2>#r^ IdtusrtSfutttfiatMboUdh' 
liqudK)€rtic€ttihabtttttjitt a» tftqttsdruflutitU^ 
sttsjtutttfmul d b , b c ifut lintu c c ,y&rr UttJtd c a 
f^uducAtur . jDmcmut tangtttt aCy&tttb ydgd 
tsct b if^dlUUtdngtnti a e iC^ a f duddturfdtdl 
leld dxi b d • Eritpcr idm dtmonfttdtd quddrdStmt 
h e . if^iirif/^ Ytiidngulo b e Ci quiidrufld ttidm dqud 
Ud tfunt ; hoc tftquddrdtum a e , wl f b , dqttdle tritrtffnm^ 
gulofub b e jxfrqttAdrufld e c > W/jv^ f a » <c^ qnddrttfU eitf^ 
Jtm cc. StidTtiffd tCy^tid db, bc ftnmlfunt qunrtdfdrs 
UtsrisrxififdrdboU oiliqtu cttius nserttxftt ^&didtttdt^ a£ 

Nos duttmcUctbMmns in fttdctdentiCo^ j^ 
roUdrio^j Untdm a b ^ qud mttitnr imfttnm 
froitSionum^fttu qudfubUmitds eBfdrtAd 
U.ohUjqud bxis/trticdm Jhdbtntisin by tftfe 
qudrtdm fdrtdm Idterosiroffidiufdem fdrdke 
Ub&^^odejfeverumcenftemMttimnsetid 

ex doSlrind AfoUonq , cum Usted a b ttonftet txAty^ exfitiU^ 
mitdtCyvel^lMdrtdjdrteUtirisre&ifArdboUrtSjt qnsvtni- 
xtmhdbet dL 





9 A^ r s 1 T i^. X X 1 r. 

Vaelibetparahola infinitas habaXuhtiociieatts. 



%i enini per f>widim (tibtime d»cpsik 
tfiperit Galifeus« liaea ^orizomalis dt produca- 
tur-i<)ua^etJinea peq)enilkuiaMs.auaE«xfaaccle 
nutaturin j>arabolainXublimitas eiuldem parabo- 
Jae crit, dttmmodo in^)etus a mobili per dercenfiioi 
4U]uifiais «onnettatiir.iion per tineam iiQdzoQta^ 
laln« fedxaiigeiuenu 




« . 






Plropofita fitparabola mSc, aiius fublimfetf sd, ikpcrJ 
j^aturhorizontiMuidiftans Je, Oetnittattirttnr qu^libet e 
*. parallclaipfi Jm, Dicor^fublimitatemcflcparabol» 4 
^r,dununodomobileinpundo k conuenienti roodo conuec 
|atur,hoceft|>crtangentemin pun<So i. Vel. Oico^raue 
poft cafum r^pcrtaBgehtem Sf. fiue ig, conuerfum.propo. 
fitamparabolampercurrere. £ft eaim idem impetus cadcn- 
tisex c in i,acvenientisex dper Min i, Cumcrgoinvtro 
quecafurepcriatur in ^ idem impetus , cadcmq,- dircaio,fiac 
veneritmobileex # in ^, fiueex /per ^.in ^. continuabie 
ihobilepereandemlineam S^f curfumfuum. £adem dicennis 
deconuerfioneper Sm poft eundemcafum ei . Qmic t^ ^ 
blimitas eft parabolae ^ii r . Qupd&c 

pRQpasiTio xxr, 

DAtisbinisquibufcunq}, fiueimpctu&diredione, fiuelm 
petu&ampliaidincuueaniplinidioc&dire^oae; fi>> 
cum parabolae reperire . , ' '^ 

loxta duo data conftruatur figura propofi 
tionum praK:edentium , & producatur db c 
donec concurrat cuoiaxe parabcffg dc . Di« . 
co c focum efle parabolas r Cum enim per :^ 
conftrudionem asquafes Smebybd^ asqua- ^ 
leseruntetiam ^r^^^rinter eafdemparalle 
lsiStSce4.dc. Scded fublimitaseft para- 
bola?/i/>ergo dc quarta pancftlaterisreAts&propterea r» 
focuscritt QK>d&c. : . ii, 

Cbrollarium. 
HincfMetf»MiU,qmtfemfeQ4tfim B^erefieim «1»^ 
rizeMf4u limedi Mimirervere fniiei$efifiam femsre&sfiemm 
MefeftAhmmitt%^fiku»t«sfMf94hiii^pi*m» > 




4M^ 



Z \ 



PRO. 



IM 



DtiinmP^eu^um 



\. 



.1 



P 



P ao P if S l T t O XXP"!' 

Arabolaproiedtionis horikontalis nuuunU oainiuia cft» 
<|uae fieri poffiat aJi> eodeoi in^tu . 




. Sit kppetus db . fiatqs drculus dJt. Sit 

etiam parabola iadus horkontalis bc^ Si 

alia parabola i e ^ Dtco maiorein efTe para«- 

bolam b c • quam i e^ £ft enim h 4 reAa fub 

loAiusparabdi^ ic^ Schd reda fublknitas 

pa^abol^^r, Quare bc omnium maxima 

erit ;>cum m^djorem habeac fublimka tenv idcoq. maius lalus re 

ctum« Qupd erat&c 

p jt p s I T j a xxri^. 

I^ Arabol^ ab eodem impem&^ar, quarQ dtrciSHapesssfaa^i' 
liter ab horitoiMe vaimq;diila« &dufi^ 
dem pxirabolf portiones funt » 

Sit impetus abSic fiant prolediones iuxtd 
dirediones bcy bd^ squalibus am^lssab 
horizonte bi vtriniqi;tdiftaiites,DiQt>para« 
bolam b e &parabolam i^. |)ort]ones ^uf 
demparaboiseefTe* Prodocamrenim^ij^. 
Demonftratum eft Piropc^doti^ 7. huius, 
qupd fi fiat proiedio cum eode impetu iuk« 
ta diredionem bc^ (iue bg^ parabolx harumproie^ofiuoB 
v)iafti eaodetnq;coDtimiatamparai3^^ efficient. £dt etgo 
b h eadem paraoola ac i ^ s quare eriam bftzAtia parabola^ 
eritac bei quandoquidem asqualiter inclinanmr iliredidnes 
/^,^^,idemq5efti2ipetus, 

•^ « ^ T o s j r To xxriii, 

ab eodCfii^paD^ ^vt^QavodeniiQipedi ^k «codeoHS* 
poris mometuo £mul jroidaiitur ^uia per diuerfas 

jocli'» 




s 




Lihr St€smdus . ist 

mcliiiadariesiurfum omnia grauiafcmper 

in periph^ria alicaius ciitulicuius ccntrum erit in perpendicu- 

Sit fada proicdio horizontalis i e^ & non 
horizontalis qua?Iibet aha 6 h iuxta direftio ^^ 
nem bd. Sumftoqjinhorizontaliparabo- 
laquouispundo ^, ducatur perpendiculum B 
rr,&hori2ontalisrefta</l Seceturq,- ^^/ ^ 
^ualis b r, & demittatur perpendiculum dh 
«qualc ipfi ce , vei bf. Dico grauia eodem 
fimultcmporccflrein ^ &in >&• Cum enim aequales fint * ^ . 
i ^•eodcm fimul tempore grauia eflTcnt in r & in ^ fi ^uabi* 
Ji motu proccderent . Scd cun^ grauiras operetur , ergnt graui- 
um<defcenfus eiufdemtemporis, ^fquales. at per fuppofitione 
deicciiAi^ vniuseft r r , crgo dcfcenfus alterius erit d h. Q^a- 
rcgrauiafimuleruntin r &in i^ . & propterea in pcriph^ria^ 
circuli , cuius ccntrum cft/, nzmfe.fh ?quales funt , cum fint 
faccxa para^d^jgnmiM^ bcybd. ^isgniz,^ 

VerMw ergo efty nenj^hm^ntend cMdeMtU db eodem fm£fo 
fer diuirfas pUi2orMm imcUnMtiones yfedetiim froirftdfenfer 
effe in eiufdem circnliferifhmrid . Exemfti ffratta ; ftqnis^ e» 
atiquofunCfo grduid ffmcertf ctm eodemimfetMferdiitfrfAS 
inclindtionesyMiudqigMt^wnfteret todem temforis inftdnit 
exquieteydrdb eodemfmnSh ; videret grduid froit&d femfer 
in dliquo circulo diffofttdcwmedte^ (jr huiufmodi circulusfem 
ferhdberet cen tpum in grdnt Hhy fuodndturdkthdrfccndi 
emiffumexfuiett^.ivt^diehmtB. '^ \ 

P R P 5 1 r J O XXIX. 

9 

l ab«odeiiipnnAD,&«tim eodem fcmpcr*^"]'?^' 

proie<fttone$ >verticesparaboIarum, Ciof^ 
oittaiccilfiwiiniptodUtrcBitinrpliaenAdis lup^i iiv^>v ^ 
..V. ius 





iusquideminaiordiamecerhorizontalisfit» &dttpfai iniiioiw* 

Sit itnpetus di^Sc circa a b • Piat ibmicir 
culus ddh . tum fiantproieiSiones iuxta tan p 
gentes dd^d^^ Dico vcxiic^ paraboloiu 
elTe iniupeificie fphan^oidis ; qua? habeata<* 
xem dh, &diametrum horizontalem dup^ 
tam axis dh . Demonftramm enim e(l Pro^* 
pofitione g. huius , quod produdtis horizon 
talibus^ perpunftum d^ & A/per puniftum ty quae duplar 
fint hnearum fdjhe^ demonftratum inquam eApunda g & 
i efle vertices parabolarum. Sed punv^ g^ i ^unt in fphae- 
roidis fup^rficie, dequadiximus (efl: enimvt gfa^fd^ ita ik 
ad A^;ergopatetpropofitum. 
Spbfrd €rg$4i£Hmtaf$s d^cendennsfroie^orurH eS infdferfi* 
cie fphfrotdis iUiusfpeciei qu( didmenum hdbedi dufUm dsiis, 

Lemma L 
SiteBdUned dudspdr^otds contingdiin eodmfnnBo^fin* 
qnefdrdholdrnm didmetri fdrdUelf^ ifjffdtdhoUfe mntno ea^ 
tingent in illo eodemfunSlo . 

Sit reffd lined ab fns infnnffo h 
dHdsfdfdbolds c b d , f b h • contingdt , 
(^ hdhednt fdtdhoU fdrdlleUs dume^ 
tros , Dico hninfmodifdTdboUsfe mnttta 
€0ntingere. Si enim non contingnntf 
fecent ; & intelligdtnr dlterdm fdrdhoU 
^nmejfe cb h .dlterdmveri f b d « jfgd 
Nr hi fdrdUeUdidmetfisyffr cd ^fdtdUiUtdstgenit . Erttnt 
V^^^^udles ci, ihiitemf^ndUs £i^id* ^god ejf imfofihi^ 

Lemma //• 

Sfdyp^Mdzhc^dhc femntnlcontingdntinh^^bd 

htdntdidmeh^sp^^lUUsiDicohdsfdrdhoUsnnmandm dmp& 
,ms €onnenire . , 

^^^^^f^l^^^f9tnnenidt^tin(.&dti€dtitthhft(rdai^ 

Udid^ 




L$$er SecunJb^ • 

^MMfis ci trdindiim frodtKdt» . Hd- ",' 
bebit er^>qiuubt4tMm fh tdmdem rdtit. * 
mem dddtte qtuuirMtd c i , e i , nemfe qitJt 

Met hhyMdhu ^deH imftjjtlile 
nge&e. 



it% 




f Jio p o s j T I XXX, 



S 



I ab eodem punao cum eodeiQ fcmper impetu proieiftto» 
nes fiant , parabol^ omnes contingent fuperficiem coaoi- 

disjparabolici.cuiuslatusredumquadniplum fit proie(ftioiiif 
iurfum j>erpeodicu2antilr.j&ds . 




Iitin^)e(us dh,ti^iKzeth fiatcircilkis ^/l^.circaqtaxcm 
eU> fiatex veftice k parabola^/^^auns focus Gx. 4 . Fiatiam 
proie^oiuxtsLi^jamlibeteleuati^mcm 4^,fmQaturG|; dewcmak 



hsif& df. &demiiroperp«idi€uIe •hei* fiat fuaparab 
ciEcadiametrum-M txpua^ 4«^€rit<]}h^p4rabola femitiu 
vroic^abiiDpetu i^f iuxtdlineamdircduiain 4d, Hvox. h-t 

wbu- 



f.c^mk. fublimitashuiusjparabol^f Ducoftir tri quaecomicoirtcwn 
parabola tlc Ccnueaiatiii/* Dicoparabolam jfpecond- 
nuatam contingerd parabolam tUiriL Dtacamor cmiina- 
tim /;» , & /r vfq,- in ^ . 

SuDtper Lcmma Propol. 24, proeccdentislibri in continua 
" ratione /«r, ofy ef\ Quadratum i/^.quadruplum eft redango- 
li ^ ^/bb parabolam cuius focuseft a. & quadratum efcxxca 
iitperconftrudionemquadruphim quadrati df quadruplum 
etiam eritre(5languli dfb . His d^monftratis ficprocedemus . 
Keda «r i ad rediam ^/Bfr4.fextieftvc mlad fe, (luevt 
quadratum/p ad quadratum/V ( fumptiiq; eorum f ubquadru^ 
plis^vtredangulum ^^/'adredangulum afi^ hoceft^^omif 
la communi aititudine ) vt re<5ta ^ ^ ad af Qoarc diuidendo 
erit vt mfadfb , ita bfzdfd^ & proptera in concinua ratione 
funt «r/*, ^/i/"^ ; fiue ne^he^ei. 

Tranfeat iam parabola d e . per pundu / . erit quadratu 4 / ad 
quadr./ niViieyzden, hooeft vtquadr. ^ >& ad quadr. e n hoc 
eft vt quadratum bh.ad quadratum ir /; & permutando , <pu* 
dratum ai ad quadratum ^ i^, erit vt quadramm /^ ad qua- 
dratum L Quare fn^nL ^uales funt . & ideo parabola 4 
^e.cum tranfeatper/.tranfitetiamper /. Sumatur tandem 
b r aequalis ipfi ^ X9 .& iungatur rl . Manifeftum eft r / vtram 
que paraboiam contingere , cum fint a?quales tam mbybr^xvi- 
ter fe • quam n e , efmiQx fe . Ergo parabolae dpe^Sicblc(t 
mutuo contingunt in pundo / per primum lenuna; neq; ampli- 
usconueniuntperfecundumlemma. Quoderat &c. . 

Sfhdrd ergo totdlis dciinitdtis proieSforfm eB in fttperfcie eo» 
noidisfdrdbolici^euiusfoctis eJifuttSittmex quofiuntproieSfio^ 
nes\ & Idtusreifum conoidis quddruplum efiproieiiioms per^ 
femUculdrisfurfum. Dtm^nRrdittmtnimfuitfinguUsfingu^ 
Utrttm proiekionum pdtdboUs huiufinodi conoidis fupcrficiem 
dtiingere , tmmqudm a ctdeve . Proieffd igitttr , eodtmtemp^ 
refuntin fphfrgfupctficie y infine^cenfionis funtinfphpr^ 
dis fttperficie ijupremd illnmnd&mtdceS incenmuUt purdtu^ 
Ueifuperficie . 

Lcm- 



-9 ^ 





ch,hf. 



dUmtiffim • a h defcrifU dfymftoujHnt ; hoc efi 

SHmfemf^mjfgisMc^ 

t9f€n^4^Mie$$i$mty' 

. Ke£fangiilumffib a d drUiere reHo diffetentid 

^ efi inter qnadrata b g , g e , item etiam inter quadr 

Ergo redanguLa etiamfub e 5 , cf ' ^^ s"^ *ar>Afi^u>a H)na tinta^^ 

fub icjclit tamquamvnalineay aqualia erunt inter fe ^ cum 

fint dijtrentia quadratartim^recipfoca ergo habebunt latera^ 

tumpevt cb ad icitaerh ch^lineaadhgc iirieam.eftau^ 

^ tem chi lineamaiorquam bge, ef ideo eb maior ei^njuaik 
£c • Parabol4 ergafemfer mdgis dccedunt . Suodnunqu4m 
ionuenianipatet. '^ 

Namfipo^biie eBy conueniantin a . ^ducatut 
ordinatim a b. Cum parabobf fint fquales habe-- 
huntidtm latusredum , eritq*^ quadratum ab • ^ 
quale vtriqi reHangulo qttod c ontinetur fub latere 
teifo^ dr Aterutraipfarum cb^db* ^odeftim^ 

poj^ile. r 

p R o p s I r 1 o XXXI. 

PArabola proiedtonis horizontalis nunquam conuenitdl 
fuperficie conoidis praecedentis propofitionis» etiam ii 
femper magis ac magis ad illud accedat . 

Sitin figura praxedentis propofitiom's impetus 
4b\ parabola genitrix conoidis Cit bc^ parabola 
autemhorizonralisproiedtiomsfit ad. Dico has 
parabotas femper quideni accederC') riunquam ca- 
men conuenire • Sunt enim circa eandem diame- 
trum ^ ^ i & funt a^quales ;quandoquidem re^a ab ttt quarta 
parslateris rediparabola?^^, per conibui^oncm, &para- 
. hola? d dy quia eft ipfius fublimitas • £rgo per lemma praece* 
densafymptotterunt. Qgf!klerdt&c 

Aa • Coro- 








V 









%t4 Dimiufr6U^^^ 

f CoroUariiiin» 

^unmn^efim ififHrdoUsfi^drUxti Sremtnes ie9f^ 

fmh intUiMtiU nmn^tiam twtingere fuperficiem contidis j At" 

. t0menfic9ntlnuiUfinteUigaHti»iUudc9ntinient adfmesf 

\ fftfitusfuferiwes , Demenfirdnimus enlm Propef. 7. & iv-Pf- 
• fnieUs diTtdionumdeorfum vergentium edsdem effe M dirt 

&ionnmf0rfum tierj^pnuum^^dummodo lintf dire£fiomtm{f»d 

fffirdkiltfriiitnftdf^entvtrhnai 

' ' ■• ' • 

P H P S I T I XXXII, 

DAtt) ipipetu fiue fublimitate dc, cuius proicaio femire- 
da fitparabola net, Dico , fi proieaio fiai cum eodcm 
impetu horizomalitd: cx punao fublimitads t , iaa«m, fiuc pa 
rabolamcaderein ^ . 

Cadat enim, fipoifibile eft, iadus 
horizontalitcr £11^ ex pundio 1: in pu 
^um d, £t quia paraboke ed impe- 
cus, fiuefublunitasponitur rc^ ne, 
crit^«- . quartapars lateris redi parabQ 
hscdi ergo dd af^licata ea fbco dup 
laeritjpfiusii/. Sedctiam4^duplaeratipfius ^rjcumfi^ 
ponatur d^ atnplitudo fa^ i proiedione femire^, ergo 
aBQuales cflent ndtdkt impoflibile • Patet ergo propofitum • 

Patetetiamquod iadus c b defcribitparabolam genitricem 
illins Conoidis , cuius fuperficiem tangunt omnes proie^O; 
nesfiidaeexpundo d cumeodemimpem, 

F RO P p S I T I XXXIII, • 

Atoimpetu, & quocunque plano fiue eredo, fiue acIIkh 

' rizonteinincljnato,reperireindatoplano remoti£Smu» 

fiuc aki£Gmuin|»Bidum ad quod cum dato impetu fieri poffit 

Itemrepenreditc^ioiKm^^«^isiWB9fiaeJ%^ 
iUnmlaflumficiat^ '" 

: sic 




^ 



llf 




IMt SiOtndSl 

Sirifhpetiis mB &parabola CQnoidis 
fit ^ r . lamdato plano ad horizo&tetti .^l^^^^v/^ k< 

credo^r«eritpundiiin ealtiffimum om 
niumillorum»adqu«poteftex 4 cum 
impem 4^« iaduspenienire. Siver6 
indinatum fieplanum vt/A, erit pun^ 
k aitiifimnm omnium iliornm ad qu«cu datt>impeca expUB^ 
do d poteftiadusperuenire. Dire^ionemver6qaa?&citpa<« 
rat>olam pertinentem ad pun^om A Gc inueniemus » Fiantcir- 
caaxem ^^circulus» &ellipfispropofit2onis iptiundaq. I^t^ 
fecetur dlipf^s in # , & ducatur i m . horizontalis qua? iecet cir« 
culumin /:erit ^/dire<^oqua? parabolamemittit tangentem 
' conoides in pun^ h . Hoc enim demonftranir in Propo£Ki<K 
jahuius. ^ ' 

ProfofiM Afchimeiis tjlfcf^nf Ummdin Uhr$ dtffhftoi^ 
dihMs & Ccnoidihns, qndm (dmcn cxf cditins dcmonBrdhimnt. 

Lemma • 



SifiicriifdtdhoU ab CyCuius hdjis a C , tdngcns % 
d yfdTdUclddidmctro c d >* ducdtsnrf. dlid fdrdUcld 
didmctro^tiiDicocffcvtcfddizyitdibdd be. 
Eficnim cAddbclongitndincvt da ddzcfotcn 
tid > vclvt cdddfe fotcntid . Snnt crgo continud 
cd^fcbe. Itcrumcfl y%;t cz dd zi itd cd dd 
f€jVcIfcddbc;(^diuidcndovtc{dd fz$itn 
fbddbe. Siuod crdt drc . 

Mdncntccddcmfigurd ^ dcmonfirdtione ^ 
dicofifroducdtur rcl^d a b vfq-y in h> (^ iuto* 
gdtur fhfquodfh, t^ zdfdrdUcld erunt. 

DcmonRrdtumcnimtB vt cfdd iz itd 
g/c fb dd}}c,hoccfi ch^^hd. ^dri 
th .f^dUcUmt iffi dk^. 



Ajt i fAO 





iSt 1 *De m9^^hmif^driik 



' ! - . f _ » 



DAta deuatione & aniplitudine patabglae In plaoo hori-« 
zontali, qimitur amplitudoinplanQ indinato« 
Sit in praecedenri figura data eleuatio jtk^ amplkudo autem 
^^lanumq^^datuniiit ah qua^riturtrai^tusparabolst^J.Du- 
cantur V ^paraUeJa diametrb>6/vcr6 parallela tangenti^y^. 
pa:raliela diametro > Dico b d^o^ tranfitum paraboke . Hoc au 
tem patetCKdenionftratis* . 

. Datain eademfigiaraeleuationc i^^ & bafi dCy pknoque 
ft ad horizoAtem ere(9:o^ qua?ritur pundum i • iii codem pla^ 
vsijt. Ducatur cd ereidla adfhorizonlem,/^ parallela tan- 
genti ^ii^i&iungatur hdMc^s efin b^ Patetiterum tran-' 
fitum parabolse elTe puni^uml^ 

T R O P V S 1 T 1 X> JTXXF. 

BAta bafiparabolae, vnicoqjpunfio per quod ipfatranfit^ 
Vel datistribuspundisinparabola, ekuationem pro- 
it(5tionisdcinonftrarev ' ^ • - 

''Sit in eadeni iigura data amplitudo i r , datumquepuniSuni 
^.. Vel^cntur tria pumaa vtounqi a^ b.c 

lungantur «rf^,4^^ fic.pcr ^,i puniai-fintparallela?diamc- 
?troir^^&/if r»,&dabuatur psxi&z h &cf. Prodiicatur crgo 
^f qua?parailela tangehtieat. an^irius ergo i&-rf eritangulus 
ndcuationis^ : \ . 

•Manentecademffgura, DatoanguIoelcuationisv&/rda- 

> «isq; pundis c &i,* inucrare punaum ex quo facta fnerit proie* 

^ctio.. Aganmrpiei:puncta>r, &ij,,^ horixonti perpcndicdlafes . 

^fy dj: y quae dabuntpuncta^ ,/1 in linea)i&/'ciitai .Ducabtur 

iiam ^i h i^ jquaftconcurrant Verbi gratiain d . Et cx puncto .4 

^cta erit proicctio aificoncurraat impolSbiiexlatamcrit^ 

Xcmma. . . 
iSihfdrdbclacuiMs Idfis c d yfdrdlUUdidmmofiterh zh^ 
^itrt&dngmkmfiib^Sihi^ cad^ 



/^ 



I^er 




*ufn£i4ngMlumt\. ^Ucdfe^deft Hi 

hifdrUm cr non bifariam , mtquddtAtH - 1— u: 

\Ayhoc eB reHdugulum e i dqudle reCH 

guU cad &quddrdto la • Demftisf^ 

qudlibus (nemfe hinc quddrdto 1 a ^fiue m b , c^/Wif reStdngu 

lo t\i)reliqitdAqudlidtrHntyboceBre£idnguU mi,c^CAd* 

^oddrcj. 

Idmfi re£fa a b , c d • fuerint pdrdlleU did- /^n^ 
jnetro , w> recidngfdtim e a iddreUdngulum e / \ 

cfv/iab,Wcd- E A Q f 

SuntenimiUd re&dnguUdqudUdreCidngu* 
iisfub ab (^UterereiiOy (^fub cd dc Utere reSto refpeCfim 
' uCy ifiavero cum hdbednt dqudlem altitudinemy eruntvtbd^- 
fes ab,cd. ^dreetidmreffdnguU e a f , c c f , erjtntvt abt 
^cd^ 

» 
I 

PltOPOSJTIO XXX Vt. 

DAtadiredione ^i^,&bafi i*i/.datacftaltim- 
do paraboI:]e ibpra quoduis pundtum r • Di- 
uidatur bifariam d d in/, & engatur/i^ . QupDi- 
am damr angulus bd d dirediDnis , & bafis dd^ 
dabitur in triangulorcdangulo latus bd^ dcidcof /^T^ 
b^ quaequidemeft.quanapars ipfius bd oh para« aics 
bolam«FiatergovtredanguIum ^/^yfiuequadra 
tum femibafis Afzd recSangdum dcd^it^ dXittyidofb . ad ali| 
am «^ & qiiarta reperta eJcitdltimdD qus^ta ct • C^od erat ^ 

« 
' t^emmd • 

Sicpnoides parabolicum dbc feceturplano d^fx(\\ndh' 
lbinteraxi»fe4^parabolaeiit, &a?qualis femper eiqua? co« 
ooides generauit , hoc eft asquale k^tus re^um liabes . Sump- 
^Ofioim^uoltbetpundo i infe^one i/i ^«applicetuc H^ du* 

4:acurq. 



J 





oXV« 


\rT^9 


w 


\ 




/ ... 


~-^ 


n 


P K « 



1^6 Di motuTrohif&mi 

caturq; ini/irparallclaad ^r.Iam:cum 

«quale fitquadratum « Jf redangulo d 

f c , erit quad« df ad quad. djk^ vt re^ 

/ r ad ^ ^ ob parabola ^^ r»fed quadi 4 ifr 

ad quad j»^ eft vt redbi&^ ad ^^>&quad» 

m0 ad re£tangulum m InfiMt ad quadi / 

eft vtreda ^t ad /<;ergo ex asquo » quad* ^//^ad //, cft vC 

rci^ pezd<l. Propterea fedio die parabola crit • 

Amplius. Qupniam vero redangiuum fubdiametro fe^ 
& iatere re^o parabol^ de/xqvalt eft quadrato appiicata^ d^ 
fiue redangulo dpe^cvi redangulo df e aaquale eft re&ang^ 
ium fub reda pe^^ laterc redo parabola; 4 ^ ^ . ( per lemma^ 
Projpof. praeced.; aBqualia erunt inter fe illa pr; dida redlango- 
la ; led aitimdo f e eadem eft vtriq;» ergo bafes arqualcs enmt» 
nempe latus re^m parabolas die^ aequaic crit iatcri rcdfco pa« 
rabolaei^^. Quod&c. 



FtibposiTio xxxrii. 



s 



fius excipiantur in aliqua fuperficie plana ad horizontem ere* 
(Qa » Dico omnes illos ia^ in quandam lineam parabolicam 
cadere ^ualem femper parabolasproiedionis • 

Hoc autem patet ex lemmate praemifTo • Nam omnes iili ia* 
Aus iiorizont»ies fuperficiem quanda^ defaibunt conoidif 
parabolici % quam fuperficiem fecat plan& iliud credum in quo 
feriunt iadus » ei^o ieaio in quam cadunt iactus » erit parabo* 
laaequaiisparabol«genitriciconoidisiProptereapatetpropo« 
iitum* 

Si vero iactus omnes terminentur ih liorizonte>fecdacircu- 
lUs erit ; quando verd in planis inclinatis» fectiones erunt elUp* 
(es , quod fiicile colligi poteft ex demonftrationibus antiquo« 

rum^uiDcmooftraueruDtobiiqu^miectioDemcoDoidis elli- 
pumcfle* 



'■'■" ZJkr SehmdM^ 191 

PEMOTV AQYARVM. 

IJm jfirh ^ de 4q$ns dUqtmm httic MeUc etnttn^Uthnm m 
ftrere moH eri$ i»e0HtifHft»s : MqMis enim fffceteris etrftri'- 
busfubUiHmAMSdde^peeMUmSiO^eognMmsvidetttrmttuSiVt 
fcri Ht/HfnJm quiefcMt . Omtte mdgmtm iiittm HittMHtis m^ris 
m§ttim i Prfteree pitimfimHemfitimiHtimt dqHdrum^i curreHtiu 
tum meHfuram^ ttim vfitm , quttrtim emnis doHriHd refertAfri» 
mumfuit t^ Ahbdte SeHedi&oCdBeUUfrAceftere mee . Scrif^ 
fit iUefcieHtidimfudm, fjr' iUdm Hdnfitlum demcHfirdtioHe, verS 
etidm efere etHfrmduit » mdximd eum VriHciftim drfofuUrum 
vtiUtdtCt mdierecttm ddmirdtionefhjtefofhotum . Extdtillius 
liier^ vere dureus . N»s minutd qtittddm , ^flerumq; inutilid » 
men tdmeHfenitus iHSuriofd circd hdnc mdteridmfrofequemttr . 
Supponimus. 
Aquds violeHthr erumfentes in iffi eruftiomis fttnB» enHde 
imfetum hahere , qutm b^^eretgrdue tdiqued , fiue iffius dqu* 
guttdVHdyfiexfufremdeiufdemdqMdftiferfciev/qi ddtrifi'. ' 
eitim eruftienis Hdttirdliiir cecidiffet , 

JExemfU grdtid . Situhut ab cenueHieHtis ed~ 
fdcitdtis , hec efi mdqna Uxttdtis, iHtelUgdturfem 
fer dquMflenus vfq\ adlihelldm a , etfe^etttr mh 
gufiwtrificioin b . Suffonimus dqudm ex b erum 
fentem^undtm iiTtfetitmhdhereyquemhdheret grd 
ue dUquedfi Hdturdtith ex zinh ceciSffet . 

Hec ratione qufddnmodt coHfirmdri foffe vide~ 
tttrndmfiddofculum b dUnstubusinftrdtttr^etex 
quift\eo'dftetur,dqMdex b iHfiueHsintuhum bc, 
tdntam vimjbahet vtfe iffdm CMehdt vfq; ad eadtm 
liheildHihmzdHtalem zzdnStdm fererificium a. 

Ms^' verifimiie vtdettrr etidmquande ipfdex 
b lihefd erumfitjidhere vimredeMdivfqiddhoriza- 
tdlemUneimtqMffer a ducitun velquodidem efl 
^dherf tdntum imfttum qudntus efigrduis dUcniut , 

ftMi 




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i 



m 



ipjr Be mom T¥oie!^mnf 

JiuevmusguttsUbericadentisex a m b. 

Experimem^um, etUm aliqti^mj^ ffiticifitt^ f^ofirttmff^ 
bdtyqudmquam dliqud expdrte refrobdre videdtttr • Ndmfi 0/irM 
tum h furfumdirigdtur\ & fitdpt} ro$undum\, c^ Uuigatmtto^ 
fttqfeliqHdtotius tubildtitudo multo Cdpdcior quam arificim h^ 
wdebimus ^qudfklienteperlinedb Cjqudfiad libei 
Idmfiidmzdi dfcedere. DefeSHonisdute c (Xcdufdm 
adfcribere pojfumus pdrtim impediment^ dcris qui 
c^dttaquodcunqicorpusTHobile lu£tdtur\pdrtim etia 
ipfimet dqudj qux dum exfdHigio c reditum dfitifdt 
deorfum^feipfdmvenientemimpe^dity ^retardat^ 
neque finit fubtuntes guttds ddillud ipfumfignum 
adquodfuo impetuperuenirenty afcetrdere.pojfe. Hoc 
manifefie pdtebity^udndo oppofitd mdnuforamen \>"penitus oc^ 
cludatundeinderetra&aquam citifjime manu repente aperia^ 
tur : videbuntur enimprima , & praeuntes gutts altius pertte^ 
ntre ^quhnfit deindejBulmen aqua c.pofiqudm dqud deorfum 
fiuerecaperit . ilU enimprioret guttapracedentem aquam noto 
halfent , qua contra ipfas refluens motum ipfarum in fine ^cet^ 
fionis impediat fuppono enim duBum b cperpendicularem. ' ' 

Addeetiamqttodfiquisobferuetaerem ipfi aqua b c firfn 
fufum^reperietipfum agitariy drfurfum mBueri^ quaquidem 
latio nonfitfineviy^ proptereaQumimpedimento motus aqua 
afcendentis ^ Vnde efi , quodfiquis velit de hocprincipio expe 
rimentumfacere ^fumendumeffetarget^um viuum^ quodobih- 
timamgrauitatem tndgis dptum eft^^dd conferjtdttdum diu co 
eeptum impetum , c^ adjuperandam aeris refiSentiam . Aqua 
dutem ob leuitatem multum dberrare videbitnr ^ ' ^ pracipuefi 
tubus magnafiterit altitudinis: tunc enin^ ob maximum impe^ 
iumfpargitur in guttulas minutiffimas tarnqtt^m rerisy neq; di^ 
ptidiam^&fortafietertiamy quartamuepartemaf^ndit illius 
interualli quodre ipfa , theoric} loquendoy & remotis Ttnpedime 
tis omnibus concepto impetu totum exaquare deberet . Cfterum 
fiquispYfdidisrationibusnonacquiefcat^ videat an interfe^ 
qnentesFropofitiones vliamprobetiqHodfiitaerit^acileperre'^ 

foUt. 



IMer Seemduf ; r^ s 

lMMm€mex4fpt$Bai4frofofiMnefrimdm/hf^Jhhkm denA 
jh^imus ifin minns totdm hdssc dfpendicem demetn dfndrnm 
wlfdknpratermistds » wlfnnditns i Ubelle eneiU^^ qnad eqni 
demUbentt^fin^conceAo^etfifdSinm exferimentnm omniditi^ 
gffntid mdgndmfdrtem feqnentinm frofofitionnm endBifbmi 
eonfirmdnit » 

His exfofitis eonfideremns dqndm recidindm in e > nemfe im 

fldnohorizdntdUdn^oferlibeUdmorificif b. ExGdUleohdbe^ 

musimfetnmdqufCddentisex c in ttdntum ejfe qudntns w-- 

here fote^ edndemdqudm ex c in c. Ergo imfetns in t Uem 

eHdoin bjfedin e imfctns fH tdmqudmgrduis cddentis ex c 

in e . vetex a m f> (iiximus enim quodfun^um c reiffd de- 

teretejfctnlihetid ad dbBrd^isimfedimentisqnd dqudm r^- 

tdrddntjergoimfetnsin b eft tdmqudmgrduis cddcntis ndsth 

rdUtereminh. 

Hisfuffofitisqudidmdemonsirdbimusdedquis erumfenti^ 
bus y qud miri cum doiirindfroie£iornm conuenire videntttr \ 

PRimnm mdnifeBum efiomnes dquds erumfentes exfordmi 
nibns tubi dUcuius ferfordti p fdrdboUs defcribere . Pri-» 
mf enimguttdfcdttnrientese tubo funt de ndtttrd froieSiornm 
qudndoquidem iffa yqudmqudmUquidd, dttdmenfnnt ffhfrtt* * 
Id grdues (^ coherentes j i^ideofordbotdm cert} defigndbnnt. 
Omnes dutemfubfequentes, qud cnm eodem imfetu emittuntuo 
(fupfonimus enim tubosfemfer dqudflenos)femitdmfrfedden 
tjnmfercurrent ; qttdte continuus iUe dqudfluentis trdhnsfd^ 
Tdhold erit . 

Obqcetfortdffe dUqnis hoc nonvideriyprffertimqudndotn^ 
biorificiumvdldedngufiumeriti&imfetusvehemens. Tunc 
enim (vft videre efi in Uned iltd dqued , quf exfbntinm fifinlic 
viotehtius erumfit) frior fdrs orbitd iUius dfeendentis mdgijt 
tenfddffdrec y et ddfdrdboldm veriiiscfinformdtd; fofierior ve^ 

TO^hoceBedqsidmdquddefcendensfereurr^tymdgisfrondyei 
vt itd d^dm y Idngnidd dtq; cnrud confficitnr . Obieiiioni ref^ 

fondetnr i nonfolnm frfcedentemfrofofitiunculdm ^fed etidm 

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^T^ / DematuFr6$iitwum 

' mamm^fhmfeqmemmm hm€ mffmnifftAtMm . Cmfii tft 
imfedimtntmm mtdy^ ^uod.adMmmiMtmm cerfmis mMHs^ud 
defemfibilim haheftd^dmem immlm^immieremfmJhm im pnie^ 
ffimihts qmf fmmt Jmat^hmisiteWcis .Sufmidemt illic mtdserid 
fi^eiiimisftmt gUti flmm^ei ifeneiftkifekemmu^rmterei \ hie 
n;eroli»eaeffy(^qHidcm.aque4. Nulliigitur mrmmfit ^qmml 
emmfumddtmemtttmhuimdeihrimff^mewriifitThmee Uque^ 
d$ y 4C imfroieSis GMilel^fruitice tamtem mmlttmmJh iffis cott^ 
temfUti$mkiu.dkerr^txxferitmstmtUyqtUtAd,h^ ivtexdffitrA 
emJdertm j^vHfieri dekerem in mtedi0 memebfimtae^ velfdltem 
gratiifjimamdterUtJfet adhibemda . .^amfuamfi^uis mtedica 
altitmSme^feUrfiqidiligetttU exferiri hfc ommi^t.velitymfitti' 
mum ^tddam ye^fterum^; im/emJ^iU deeffi ^emferiet. £xfe 
rimtemtmtttyqmedxoiis^tmfirmatiit hasfeti^\emttues0ecmUeimM^ 
culaSffaSiifuittubo qmdam^ocaffMU^iteMeUjifeda^ttim 
attitmdo f4^umXleometrictnm^xceeUbat ^ittsbafisnmoftAmo 
quadraiommceratmtimer .J^eramitta .ven ^eramt^xaeitttd^ circtt^ 
loaue humanafttfiUf mdiora \nonferferdmfaciayfedfolertijji'' 
meexcamdtmniameUis cufreUj tenuihusy e^ ad herizMttt cm 
er9cHs:^Jlqnaenim woUttter ertttttfemsfetttfercUreBiotte exit 
ferfemdiemlariddiUmdflamttm ex quo-ertmnfit , ideeq^fiehat vt 
m^fionesmoMri tuhimrit^mtaies effettt.. 

DAto tubo^i feiiif>e^pleao,&^c petfoc^^ 
c de.^oc ^&QUX^^ntii^arxa^^ dudus 

hofkontali^hoceft m tenuilamella planapendurularLDatoq; 
]it>raKmte:<]udlibet^^,inuenire ampiitudin^ vniufeaiufq;para« 
.bola^.Fiat circa ab diametru feniicicculjis^ i&^.Eritq; paraboljs 
'flizeds,ex<^ aroplitudo dupla^linea? >ei qi» horixontaliter daci- 
turin femicirctflo. EtamplKudopra* 
lR>larerumpemis ex d erit dupta lineas 
.jdh. 'Ethocprobaturyquiacuntaquaiit 
velutprdieiftum quoddaTD,*(Rq; ( per &p 
;pbfitiiin>ipftu5pun^fublime.4^> erunc 
rper Propofitiohem 5 . Galilei , femiiles 
;«mplitudiottm medio loce proponioitt^ 

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fesjottrfiibliiiutateai* & alcttudJiiani^iiacercioifliSAinpliMi. 

CoroUana. 
. Wute nutmfti^umtfitpmdfifMbms ab f^rum m d /»». 

qttiUihefidiaeiuiift.^ \ 

For4miH4 vnb ftta aqitjliter }fti/iih medte <L dtfidn* »/^ 
eet^fitdiesdmptitttiiiiies/keere, 

Manifeftumetidm eS inferieves fersAeUs femifer fifetimi* 
bits mdiores ejfe y etim habeant mdieremfnblimitMtemt bee eft 
maius Utusre&Mm^ eSf enimfitblimMs qturiM fHN ItttrieH» 
SiiiVtoJienfumeft, , '■ 

Atodolio, (tae tubo ub quodapccb 

perforatum iit in e , & emiffionera £t- 

ciat e d. Inuenienda ftt aqu^ in tubo laten- 

ris libetf» horizontalis , fiue fuperfides 

fufMretna* 

Sithorizon df, & producatur r ^ in/, 
& fecettH-biiariani//in r,fiatq;vt f/al- 
tiaido , ad/tf . feoiibaiim, ita/r , ad alnin « 
quaeeritfublimitas ^^. PatebitergoUbel 
lamaqweintubolatentisctifeperpun^um j 

^ltubos Ji^apt^perforetarvbkuiiqjin i 

i3 aquaeconi red»iguIifaperficieincoiitfflget>cuiii$axti 
lit^nibus,veftexvet>ofi&ifla<p«rlibella» T 

Sitangulusconi burr femire^us , & 

db mbusyhoceft Iiiieji<eft in qoafuntfb- 
ramiDa^potiacuraxiseoni. SnmamrKu. 
qualis edip&ed. I^acaniiq; faorizoa 
ttdis «/e.Dicapaxab^amtnm&eper/. 
Si-Mimpotefty tranfeatper b* Scaaa 
aqucfublimitas GteM, erit femiflis linee - 

Bb 3 bdm-! 




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jpS D&nMuVrote&otMm 

h^ meiiiapropoiti^Dalisincerduasflequales^r» ^iTt&prcK 
ptcreatota i/^.arqUalisedtipii dd^vtX i/r.quod eft abfiir» 
duni . Si ergp parabola tranfitperpundum t. ipfa r^^ tangem 
-eft , cum sequales ixxiidc^cd. 

Htnc manifiHum eji ^n^dfitubus in 0m$nk(sfnisfun£hs ^ 
tiffrfirdtusfuerit^ omnes endjj$$nts ^uodsmmodo oonffirare 
"vidiiuntur Adformanddm coni niiAngulifftcim^ Si vcio n^so 
tuhus , fedffhsruU in virtice iffiusfofitd , afte ferforntafit im 
^mnibusJuisfunBisy emiffiones omnes cuiH/dam conoidisfurs* 
BpUciimdginem tonfirmabunt , exProfof 3 . huius . 

AQuanim ex tubo db perforaio erumpentium vclocita- 
tcs fun^ vt lineae in parabola applicate ad fuam vniufcu« 
iufq; fubiimitatiem . 

Sittubus 4^ femperaquapleiius,&ex 
fbraminibus r^i/ erumpantfluentes lineae/ 
defcriptaq;paraboIa defciccz axem db 
ducifhtur ordinatim coydf. ;3Erit. er^o v e- 
locitasin c advelocitatemin4^,vtimpetus 
grauis cadentis ex d in < • ad impetum gra- 
uis cadcntis ex d in 1/, nempe vt i^ <, ad df. 
ex demonftrads in primo libro de motu . 

, CQrolIariam . 

SincftqmturtxdominaAbhdtisCAflellyqMdntitAtiMimt 
» €9tetMtisfer$BiMm CddqitdtititatedqHfiaceMntisfer d (^«i 
dfiJwamiuA/kmnt fquAUa)efe vt c tyod d ijiec eflofuas erB 
fentes exfvraminihiif 4tq«aUlms efle in/t^dt^Urati»nefiiU$^ 
mitattm , flneahitttditmm/narmn .» rxriiatem hniiis C^reUa* 
9ijfrimiis9mniiimexfenment9indagamt,frmdi^s. vtr^ dq^. 
htmst/cienty/f, «mmkttswnMm^^y^e^ MagiotQis , ^v<r 
witatemnoflramexitnsffkciti^ec^nflrmdnit. 

^andfi verfifwamtna in^iiaUa trmttt-^namtitatOMpm 

^teetmtiseemfefltamr^tivaem^MkthimtiitkwiititMevcUmtatMi 
<^*9ratifinepJimim*> .: . 




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IMer Seimius i, 

Slnibus <#^^cylindriciiStiiuepriiniaiicus 
perforatus in iiindo b fluat» neque alius 
humorfuperiDfundanir,vdocitates fupre* 
insfuperficiei humorislatemis decreicent 
cum eadem radone , qua decrefcunt etiam 
Jincxordinatimapplicatxbparabola bdt 
quaeaxemhabeat b4 venicem ver6 k, 
Hoc manifeftum efl . Nam quando aquis 
fumma fuperficiescrit <-, velocitas crit fd 
& quando fummafuperfideserit /« velocitas erit </. ex lam 
d emonftratis i & hoc modo femper . 

CViufinodifitfolidumabaquis cademibus coBfivmatum 
inueftigare» 

Sit vas aqua ieiDper plennm -i* ^ aD> 
{dilSmum * cuius foramenin iimdo cir- 
culare fit t d> foliduin autem aquc cx 
eofluentisfitr«/^,&folid(axisfit/l^. 
Dic^ lineafii <^ff/ folidibuiusgeniixi-J 
cenitalemefle^Vtnumrrusbiqiiadra- • 
tusdiametri /^.adbiquadFacum dia- 
tftetri 0/. fft redprocevt2^itudo/lft.ad 
akitudineQiyi. < ' . 

Oit^ndit AbbasCaA^iusietAimefthr^ad^^icmem #^ 
elTe reciproci yi velocita^n »/, ad vetodiafcm med. dWnpc 
vt itiad im inparabela/At/. Hi5^«mtflis.C^adritu^(fc' 
merusdiametri ed adquadiatum ^f eil ytcise^U^ sdy Jtddr 
culum-tf/.nempevt A/aci«w.^<IumerusautemqttadratUKex 
j^/adquadratumcx im cilvtit/ad/i^.eFgobiquadraB^ni» 
merus diametri c d. ai^faiquacfanttnm «/ eit rtcfproefi vc aicL~ 
ludo/5, adaltitudinenay«,iC^ediflcc.. .-■; .V ■,■;.',■- 

Data fit eadem figuraalti^clis^/i a po.fk >Miifatac[;fii>^lt' 
nieterforaminis fd 5 e.Qu|i:iajrqQ5tafiilurafi»f^)jKitd4nwi> 
ler ff . Fiat \ifh ad/i,acmpevt j6o.^djeo.iiamBncn» 
biqDai<h-ati»<jklcn£ta'iT<^»iien^etfi'^«ca^adalium,.({iw4'i:i 
39otfc^o i> jdcto^rii BDmenuliqaadiaQis di^mettKaf c&er 




go abea€Sctrahaturf«iJtx{>k^draUiprcraeiU^ 

decimis proxim^ . taat&o» or^d |proaitticiaibinias e£fe diafttc* 

mim 0^. 

DAta / 4/ diredtioae £ftiifar d$. 
& pufldo i' inquod Jodditaqua 
fiuens » inuenire fummam latentis aqiif 
libeilam ^ fiiie fuperfickm • Producatur 
dtd^&titf erigatur perpcrf dkultsm ^ . ^ 
^. Deinde fiat vt ^W» ad ^^» ita ^/^ad 
aliam)CUus<|uanapars6t ^r. Dkd 
per e tranfire libellam aqwe latentis 
Aipremam • £(1 enim hJ an|en$parabol:c » & «^^ paralle- 
ladiametrofergoquadratum bd aequale eritredkangula.fub 
cdjic lafere redto ^ quare repertaiUaitnca ^cuius quartampar 
tentpofUimus he) latus reAum edt»& h / fublimitas . QjkkI&c. 
iiemfnksUeram adlm «r e$<ptfimeau cum demnfirdik'» 
mhus €^ouffudni\ ^lfbdfifdmen h^^et effe iif lamell4rteHm^& 
fflaudy idqudmferfendk<eddmjfitre&4.bdi. KeUquum "uer^ 
iuterieristubi ba &c. utfqi di mtmm uqufdul^uf ^deket ef^ 
fe cdfdcijfimum ; qu$ euimiduiut eHt » tb c^dSfiuf txferimeu* 
tum euddet . ^otiefcukqi Huttm dqtsdpet ttibum Idteutem de* 
fitrreufperdngufiids trdnfire debuerit/dlfdomnid referieutmi 
^emddm^dHftt dcefde^iiim^fi fru ^imie imfetUy dfpidfid^ 
thnAfqi m^dtfi^ inHnui^tmfln\fwemdiffergdturB 

DAtadire^onQi^»t(il>i^ ^iie£« 
fluif ^i^&puni^o ^ inquodin* 
cidat aquae emiffio. , totam parabolan» 
aquf fiue&tisdefcri^re# . v ; 

iWucattiri^^^&crigiiturperpeat \ 
diculum cd. Deinde conneidatur #a 
Ductoon' ikmtredipe^^ d ty effh^ qua-« 
nimprimafitestaag^^ vtcunque^fe* 
cundaparallelaitangenti, tertiitparalle* 
la diametro . & pun^ii k>. erifr trati&us parabolf » vt conftatex 

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POfitovafe dh iiuc cylindrico, (iue prif^ 
matico quod in fimdo perforatum fit 
foramine b . Velocitas aquas exeuntispc b 
velocitad libeila?> fiue fupremds fuperficiei 
defcendentisiuyafe^ femper eadeni ratio^ 
ne refpondebit , . 

.QjandolibellaAquarinvafeeft 4dj Cit 
velocicas a^ aquje e^ceuntis per ^«Tumfiat, 
vt fciJlio 4d vafis,, adXedionem orificij t , 
ita 4c ^d/ie. £ritq^perdp<3trinam<CafteUi|^ipia 4e. velo» 
citas libellas d4 in defcedendo . lamcirca 4m diametrum £• 
antper r,& f^dnvfOLrabolxmf^me. 

Confidereturdeindealiaiibella/^. Q^ando /S. libella 
erit,tunc perdemonftrataeritvelocitasin b vtlinea i&/. Scd 
vclocitasin b advelocitatemIibelIa?/*^eritper doiJlrinaCa- 
ftellij vtfeiaio/6 adle^oncm i,nempe vt ^4 ad 4e & fic 
fempW^ Q^reveloci^aqu^eexeuntlsadvelocitatem tibel- 
l^ defcendentis iiiquQCunq; loco confideretur, fempcr erit vt 
linea applicata in maiorlparabf ad applicatam in minori ;' ho£ 
eft in eadem femperratione . 

jflirer etUm oftendetift idem huc mede . lBteUig4fffr i» vdh 
/2r a b j(]U4hhet/tclio f h, efu^ non fit fumm^fupeyficies ; (it 4tt^ 
temfHmm^fffferficiMS jd a^ J4m ; smn e4dem quantitas aqua 
tr4njeatferftffi§mm h^ftr th^.eritvetocit4sin b 4d w- 
locit4temin fh recifrocevtfeffiQ(\\4dfcffionemhifedfum*. . 
ftA feSio f h 4queielox eR 4cfufrem4fufitrficies d a {cum v4f 
fonAturcylindrus^fiueffrifm4) ergovelocitasin b ^dveloci* 
t4temfufrem4fufe^iei d a defctndentis in vafefemfer e4m 
dem rationereffondiHfi tumimmfemfer erit vtfekio V4fis 44 
fe£lionemfor4minis h . 



'CoroUarium. 
Brgt qu4ndo 4ltitddinesinv4fe ertint a m » h m > erit v4e^ 
xitds 4qu4 exetmtis tx b fofit4 4kttudin€ zm^4d sueIocit4t^ 
.oxetmtis^ex b fofit4 akkmlitoet h myV$ eJ^vttocit4sfnmmafft^ 



449 De mofH Fme^oram 

fcffciei d a ddve^iutamfumms/kferfidei f h. b$c emimfd^ 
tet . NdmptmftdfHferion concLuJkHefermutando tantttm de^ 
ducitttrhocCinrollitritim..' 

QVantitates aquarum ab eodefn , (iue ab asqualibus f om-' 
minibus eruinpentium eodem tempore» funt inter fc 
^ infubduplicatarationealtitudinum. ' 
Sit vas d b pra^cedcntis figura; perforatum in ^ . & aliquin- 
do maneat femper plenum vfque ad (igt\um ^^r; aliquando 
vcro vfqueadyv&. Dico quantic^tem aquae exeuntis quando 
altit. eft d myzd quantitatem aquf exeuntis quando altit. (^dAm 
( intellige femper eodem tempore^ e(fe in fubduplicara ratione 
aldtudinumii^ ad^m, Mempe vtrcvlia ^r ad >^/. Nam 
quando altimdines funt dm^ScAm^ Velocitates in i func ex 
CoroU. pra^ced- vt velocitas fumm? fuperficici dd , ad veloct- 
tatem fummae fuperficiei/>& ; fiue vt applicata ^^ ^ ad A / . Ergo 
quantitatesaquarumerumpentiuexeodemforamine ^ erunt 
vt^^ ad ^/»nempeinfubdupiicata ratione aitimdinum dm. 
mhm* 

Hxc fpecuiado conuenit exa^ifime cum experimeto k no- 
bis fumma cum diligentia fai^o ^ 

QVoddam vas cuiusfummitas d perfora- 
mmeft foramine b itavtfuperinAue 
^ te quodam aqu^ du^u ia d feinpcr 
pienum permaneat . Quaeritur , quo foramine 
. perforari debeat in c vt eadem fuperinfluete 
aquaplenum pr^ise ficutantea permanie^. 
Sumamr inter db^ac media d i • Fiatq; vt al« 
timdo c d ad mediam a i ita ofculum k ad pfcu 
ium c « Erit ergo ofculum t . ad ofculum c^ vt appUcata c e » 
ad appiicatam & d\ hoc eft Vt velocitas foraminis e ad veloci- 
tatem h reciproce; -Proptera eadem quantitas aqu? efflaet per 
vtrumq» ofculum b & )r >propofitumq; vas femper plenum ma« 
nebit. 

^odddmverovds ^hcttmperfirdtMmfit infundo fordmi^ 
nehffdferinfluentequoddm ddte di^nddnSin d y fUnum fer^ 

mdttet 




_j 



r^f 






JOl 



D 



n 






B 



f> idem H^iAPingtrtndd 4xih^c vt refUdtw vfq\ dd 
fignum 3L\ ^umdinrineer a b » b c . medid h c /^- 
dfqy vech ddhsL, ita qHAntitds dqud datd d dd 
dlidmquMntitMtemiqwdingefiaomnin)^ vdsreple^ 
bit vfqy adjignum a» neq; illudexcedet. ^od 
cummultis alijs huius genetisfacile demonihratur 
e\Prucedentibus . 

\ Qijarumflucmiutn /"qu? tamenaliquo vafe excipi pof- 
JLjL fint)proportionem<liccre> fine vlla temporis , veloci* 
tatis 5 (e(5kionirq; nfleofura . 

^umatur vt in prsecedenti figura, quodcunque uas ^r ^ > cuiuf 
cunq; figura? Qt , ita tamen perforaram in flindo , ut minor ex 
datis aquis fl^entibus ingefta non etfluat ftatim tota^ fed incre- 
fcat, & aiiquaiti altitudinem fxcizt in uafe» puta altitudinem> c 
&deinde non crefcatamplius ; (ed tantunfaqua^prorfus uas e« 
mittat , quantum recipit • Maior uer6 aquae quantitas altitudi-- 
nemfaciat ai. Patet ex prascedentibus aquam maiorem ad 
minorem eflfe in fubduplicata ratione reda? ^r^ ad ^r. Kam 
cumutraque^quatranfeatpereandemfe(^ionem i^ & altera 
earum aititudinem habeat ^^alrerauero cbf erunt uelocita* 
tcs aquanim per didam fedionemexeuntium in iubduplicata 
racione ai^ adcb. Ergo & quantitates aquarum fluentium e« 
nintinibbduplicataratione feidarum altitudinum ah^tc. 

Lemms • 
Sb diameteraUcuiusfaratola zh^& moiile aU 
fm^moueaturper ab ealege^vtinquocunq;pun-^ 
£lo linef a b confidereturfemperimpetus eiusfit ui 
Unea ordinaHm ex HJo funSio intra aUquam para^ 
Mam applicata « Dico bunc motum eundem ejfe ac 
grauium naturaliter cadentium . Intekigatur enim aliquod 
graue moueriex a m b motunaturaliteraccelerato^ (^conti-- 
piatur eius momentum ciu/modi ut tamgraue quam etiam moii 
Ufimulditnifptex »• ctdmtcmpore peruettiant ad punHum b 

v:c.:/ Cc pdM 




^¥o2 






\ 



fdmamhfMm mfobiUam vpi$m 4fq\ eundenifiitmmBeffe 
tum^ HdminqMQunqifurMeUne* ^h cwfi4ef.etut ^etu. 
trum diSlerumJhemAUe.JiuegjrAueyeundem imfetum h^ekit 
4C dlttrum , quarefmter etidm trdnjibuntffdtium a b , purtef 
que tffius. & hoe uerum ctiam eritf mebile mouedtur exhmz^ 
nencrefcen$e^feddecrefcenteimpetu. ^ 

VAfa cylindrica fiuc Prifinatica in fiindo perforata «a Icgc 
exhauriuntur,utdittifototo£empore inpartes aequales» 
eniiflio ultimi temporis fit ut unum, emiffio autempenultiniite* 
poris (it ut 3 • antepenultimi temporis ut 5 • & lic deinceps ut ns 
meri impares ab unitate • 

Sit uas ut pofitum eft ; perforatum in fiui^ 
do , ipHq; adfcribatur parabola e cL lam de- 
Jhonftrauimus flu^ente ex fundo aqua, iibel- 
^lam d e ita. defcendere ut femper uelocitas 
.ipfius (it ut linea (ibi reipondens in parabo* 
jla y nempe impetus in r ^ , (it ut i^ «f , in/& (it 
«ut i^ / , & fic fe mper ; erit ergo motus libellae 
e d tamquam motus deficiens grauium fur- 
vfum reflexorum > fiue proieAorum ; & diui« 
fotototemporeemiflionisinpanesagqualeSf erit fpatimn ic 
. decurfum a libella ulpmo tempore , ut unum ; fpatium autem i 
9 ut tres,&^4utquinque« Nam,exieaunateprfmi(fo,nio« 
ius libcUa^ 4 r eft tamquam motus grauium non cadendum^ fed 
furfumperpendiculariter proiedorum(quodidemeft) ergo 
motus libellft de eadehi fpatia tran(ibirtemporibus ^alibus. 




atq; graue aliquod furfum proiedum » iicmpe 
'unum , penultimo tria;& (ic deiQceps ^ 

^ifidt uds <eneiddk:fardbeUctimyeuitttMnsfit 
a b, firferfordtumjitinfund^ hmdertfottritemif 
^fioeiuseiufmodiutm^otuffufrimdfuferficieidefie 
dentis 3 dqudhjdisfit «* ^r efi ut dqudkhus Tstemfnri^ 
i/us^qudles dhitudinemoles^pehdkridntttr^y ^uotk^ 
4dme»fd^tfi . ^Sutit^mm ^opoidiit^MnhoiicM 




^rr 



' Zifer Secmdui l ^o i 

interftut quddfdtd dxium yfiut dtitudiuum • Si trgo diuid 4« 
mus tdtdm a b inpdrtes dqudieSy mt CMoides ch ut uuum^ f^ 
db ut qudtuor ;iffumq; eb ut g.fjrficdeincefsfemperutM^ 
meriquddrm. Erunt ergo couoides ch ut unum y differeniid 
Mitem c d uttridy dc utj.cf^ ut^. &fic de incopi dijferentis 
urunt ut numeri di unitdte impares . Stjfdt e uidebitur dlicui 
^uodfinguid huiufmodi dtfferentif dqudlibus temporibus exdsi^ 
riri debednt per H dtmonjhrdtdin prdcedentiifed quonidmin 
huiufmodi Udcudtioue plurimirefert cuius fi^urdfii ipfum uds^ 
4tbfiflutifdlfumhoc.efiepronHntidmus\ demonifrdtionemq. unuf 
^uifq.coiligerepoterit ex his qudfhquuntur . 

EStouasirfcgulare db cde^ct(otQX\Xxti . 
m fundo foraminc r ; & confidereri • 
tur du2e ipfius fedionos dCybd. DicO ue- 
lc^ciratem fumma? fuperficiei aquae defceti- 
demis , quando erit ^^ ^ , ad veiocitatem fu 
perficiei, auando eric ^V, rationem habe 
re compoutam ex ratione fubduplicata al*- 
timdinu/V ad r^, & reciproca Ie(5tionum, 
nempe fedionis bd adde. Concipiatur eniiirfupcr Jbafi fedto 
nis de qu^cunq. ilia fit,uas prifmaticum dime cuius altitudo 
fit/(p. lam veiocitas fedionis prifmatica? ^ ^ ad ;» ^ erit otreda 
ficadch mediani int?r akitudines. Velocitas uero fedionis n 
od uelocitatem fe^onis bd^ cumeandem altitudinem habe« 
ant , eft reciprocd ut fedio ^ ^ ad ^ ^ • Ergo patet quod ratio 
deiocitatisfedUonis^r aduelocitatemfe(^onis bd compo- 
niturexrationcreftae/ir ad ^^,&exrationefedionis ^^ad 
»^,fiue ^^ad de. 

Hinc mdnifefium efi quodnuperde ConoidepdTdbolico dice^ 
bdmus , nempe motumfupremdfuperficiei defcendentis non rf^ 
fe sq^dhilemfed fubinde dcctlerdtum . ^d uerh rdtione dcce» 
lereturi&qud rMione Udrientur uelooitdtes fupremd fuperfi^ 
ciei dqufdefcendentisinfphfrdperfirdtd,fphdro$de,dtq.dl^s 
Mdffkttj TtguUoribtts^dfill tx ^ontepldtionejprfcedentipdtebit . 

Cc a pfi TA^ 




/ . 



4o# Def^^^s^^rm 



i> E r jjtr L j s, , 






K'i 



SEquuntur Tabula^non quidetn do&is calculi vigUijs el^ 
boratae » vt a Galileo fadum' eft ^ fed ex ipifa Tabula finuu 
ac Tangentium facili breuiq; negotiotj[;aoicripta?. C^uocunq» 
tamen modo coUedas fiierint»'non minq$ augent Galilei glo- 
mm^quamlaboremnoArumcomminuerint, Cu|u$ enim in- 
duftria? tanta folertia eft , vtperinnumeras multipiicaitionums 
diuifionum > & radicum ambages^adeofdem pene numeros ap» 
pellerepotuerit» auos exTabuIa defumere nobis conceflfum 
hitf PraKii^mnocvolo,nosfQpponere vpluifife eandenui 
maximamamplimdinemfemiparabolarum cum Galileo par« 
tium xoooo. itemmaximamaltimdinem partium xoooo. vt 
€(dem omnino Tabula? euaderent > & aliqua interdum ditferen 
lia inter illius numeros & noftros appareret« Ideo in folum la- 
borem biffedtionum iqcidimus • Si vero fuppofitionem varia- 
re^hoceftnumerumhuncduplum xoooo. fupponere voIuiC- 
femus , tunc integi;^? Tabula? diuerff quidem euafifTent a Gali* 
leiTabuIisyfed nuineri poterant fine vlla biife^oneex finibu^ 
^ Tangentibusprbutibileguntur cnutuari^ 



«. i 



L 



• '*' 



f • 



Tabu^ 






fdSarum» Suppofipa nuixima amplitudine partium loooo. Suntau» 
tem numeri T aiuUfinus reSiarcuum eleuationis duplorum » 

,r 



GRAn. 

jl Eleuat 



k w 



Amplicudo 
lemipar.. 



oooo 




5 l 173^ 



CRAtXr 
filettar. 



j 



IGRAD lAffipliciido 
Eieuat, jSemipar. 



90 I 

8P 



6 

7 



»079 
2419 



g 



275d 
3090 



II 



I 



12 

l »4 
«5 



34» o 

4067 
4_384 

4695 



£l 

S4 

82" 
81 

8^ 

z? 

77 

t76 

75 



itf 
18" 



529^ 74 



20 






a2 



5592 

5^70 
tfi57 



^^91 



«P47 I *8 



«» 



73 

7« 

II 
70 
tf9 






"I % 



1 




GRAD. I 
Bleuat. I 



8829 f 59 



8988 

9IJ5 
9272 



58 



9397 
9JII 

9613 

9703 




57 
5tf 

55 
54 



97*- 
9848 [50 



9903 
9945 

997^ 
9994 f 4<5 



53 
5»' 



49 
48 



'-♦ 



looool 45 



i 



t I 



20$ 



De motu Tme^arum 



iriOiiimtts maximamaldcudiactn omniuqi proie(5tionujttL. 
Jt ab codem impctu faAarumcflrepartiumj loooo. Poni 
musergo infubieAa figura lineam^^^ effe xoooo^partmim 
Dato deinde^angulo ^idc cleuatiohis gn 40. qu^riturqtiaa 
ta£t^titMdo ^r refpe<^uipfius 4th quasefl loooo» 

r 1 




Daturquidem^/ 82^4.ex Tabula fintnim^ cum fitlinus 
verfusarcus dc gr/8o;quiari:usdupluseftanguIieleuationis 
ddc. SeddatusnumerusTeda; Me^iSj^ erit refpedu fcmi- 
diamctri^ qu^ fitpartium r 0600. Cum ver6 nos ponamus to- 
tamdiametrum dt eife partium loooo. tunc de crit 41^2« 
hocefttantummodofemiflis illius numeri exTabuIa finuum 
yceforuniexccrpti* 

Pfdxis^ 
DupBcetur data eleuatio; illiufq. finus verfi femiifis accipia^ 
tiir i & fic habebis numpros Tabulx proecedentis » qui altitudi^ 
nesparabolarumifiueproie^onumiQcqunturi^ ' 






. 1 



» » 



ZX^ 



«V* 



Uitr Seamtbut 



»«•« 




DecUrdtioSeqiiennsttilhMUi 

PiOiuoms^i amplii;udinemoii^ - • 
fetiiiparabolar& eflfe p^u xoootf 
I)ata iam eleudctofie cnd "gr. 3 o.mxti 
quamdirigendum eft tormentmn. Quas 
riturparabolje dh Aitimdo»&fublimi 
tas. : 

i S^c^QV" ^^ bifariamin c^ erigatur 
qli6^ aBgulosre^^ c-d .Fiatq* ai^ 
lits^^^re^usJVfanifeftum eft circa^dia, 
nietniti^ a/ deicribi femicirculum qui tranfeat per d cuaJ 
redns tirangulusad i/tlonec concorrat cunr diredione # ^ « 
Cum ergo fiat proiediio cuiii irapetu/>, & dire^one d d^ erit 
amplitndo fetniparabotie liiiea dapEa ipfius e ^nempeipfa dhi 
al^oiuIoVoro / h \ y cr>/ > fublunitas tf. Qgaeritujr eigb quan* 
tit^sliaearum ^ ^ , </■. 

Cuni tfiiitpattium^ loooo. erit r^nempe femiffis ipfins» 
partium 1 f 00. femper > qusecunque fit cieuatio • Si ergo ed 
fitfintistomsieritaltitudo ed 5774.tangens aoguli eleuatio-^ 
nis r4^/,hoceft edd. Sublimitasvero ef^ii 17^10« tan« 

;ens com[]Jementi eiuickm anguli . Harc autdn veraf unt qua* 
\q ed fit ioooo.fed(^aincarunoftro edt^ tanmmmodo 

J>aitium 2500, nempe femiffis finus totius', eruht ed , efitvcoi 
es di^arum tangentium ; bpc eft altimdo ^f^iSS^.ipfa ver6 
fi^Umitas ef S 55o.propterea tormenmm illud , quod eleua^ 
bkiir ^ij 0. ^d hoc vt ifacia(| amplitud jnem femiparabolg par- 
tiuQi 10^90. cHehebit habere fubliipita^m \ fiue impetu 8 660. 
Q|ip4qukieiii4demeft^aclfidicei«naus-^^ impetus proie* 
ddoiustanimsffsedebet, qktantus efi<grau)s alicuius naturali- 
t^cadenusab altitudine %66o^ eanundeifi partium. Aititu^ 
doverdfalisparaboIjeritiSSj. ^ 

Pro^^ltkudnl^^^ 

iftoifublimitatetii^ ^omplementi ab» 

jgulieieuaaoais* pd J^u: ; 



««-A. 



ThFula comintHS JlUUidines \ & fuUimiaiefSem^ ^ 

rum ampUtudints ^gMoles fint. TaTtiumplicetJemfer loooo. Smnt 




itdajUHtftttifflrTdntmiitm cmplemenforum eleuatioms . 






fT!n^ 



fikait. .iTyDOL Imit. 
47 I 5 J«a 




o u»fm/4i\ oo 

T46»igg 



mmm mh i$4im imp€tu fsSnmm . Si9$pP0iHiM^.mmfimm dmsii* » 
fihi impnMt msxtmmt iffi loodo* Sumt sufm mmmmi 

TmkmUfimut itiSi iliustismmtm t '* 




Jjkr Secmiii . 211 



£xfd ea ^quae acfno&ftrata filiiin fropofitiombiis t /. &: 9 1 . 
!^ole^K»ratt;qi^do ^^lfUerirmasdrnadtirnjcryfiueiiu- 
iiis im|>etus ^4 horcsontem comparatus , erit 4 r ihtercepta 
•; Jn femicircUlo) durado i fiire on^^ eieuadonis* JrV.^ld &ori- 
iiowcracappaa^f..",, ■•■S' 







EF" 



(• ♦ . • 

1 >; Sttpponipius 4 1 <e0r partitmi i oooo.;nen^>e fiiiiifn tomnfi « 
'&rdatoansDlotIcnationis r^r gr.^o.quaaimus mc^ 

Manifeuume^» fa^oquaklrante i^4^ quando 4^faerit (!• 
\ faustotus, tunc cd efle finum reftum deuationis ^hoc e(l angu^ 



l^^^^f^iqu^ ^^ gqi£lis'ciIip(ri/?'.\Vltoft^ 

• |Qusjw.|>ra;ced?i» ^S^arciisfitgh^db^e- 
I jfia^ir^ fiik>\u jkOadLVfcfpelluipfius 4^ quo^efti c>oo6.[loc 

lauitem fignificat»:qii6dimpems grautsiutu^ 

j^ in i« adimpetum fiafabolae ia extrcmop^n<fto :( drnnm^do 

i a d h of |2 b nt e m ^tantam^j^omj) ar et u f ) tt ^ vt-Mo^o^od 5 t^o^ 

; : ]1B>urkdp xctbi fiUe tetilpiis lationis -Hatul-ati^ p«f ijeipriMiidUlu 

■ i ''i(^^ *^ierifi^<, (kie duiWioiidti^pafiabolacttit vf eactem |( 4 ad 

;;:4?riiempe^ru^^^ ^(k^r \ z ' : — ! 

• ' " '-^ •'• '^"^^' iY>jr//;'" •' • '■•* ■ 

$^e ipfof i^liws ve4iios ele)i^n9P§, ^ luioe b^ nxw 

'* * • , i i 4 . » - 



titiak c mi»i n tattA!t(ciuMiiiiuMfi^ieri,n>n 4 m t t iii 





; 


10 17 «»■ 4J 

t» 34 :»»'** 


■ 


30 


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88. J.I 




5» 

60 


1. ><s 
■• 43 


88. }4 

88. I, 




70 
80 


>. 
i. 18 

»• 3> 

»■ 5» 


8». < 

87. 4' 


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»0 
100 


«?• «! 
87. 1 


^ 


110 
110 


3' 9 
3^ »7 


8«. ji 


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140 


3- 44 

4- ■ 


X. 11 
8J. J! 




■ }€> 

160 


4. 1» 
4. 3« 


8}. 41 
,8j. .. 




180 


+ 54 
}. II 


8}. . 
84. 4) 


190 

too 


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8*. j. 
84, i, 




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«. 11 


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Mtfdm mubms,&m**iwiam fnkSfhMmpommiu fart, 4000. 
jw^ iGRAD. rGoaapir|i 



15. 20] 74. 40 
15. 40 







«^- 4M73» ^9 




II. 49)68. 21 

22« ijUy» 47l 



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23. t. 



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23. 52 

ra4* 18 




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770 



780 
790 



800 

L8f0 



24. 44 

25. 1 1 



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2tf. 6 



2<5. 34 

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830 



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85« 



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28. 3 



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890 



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29. 6 



29. 39 

30.' f^* 




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I64. 49 



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63. 2^ 
62. 57 



62. 27 
61. 57 



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910 hx. 43 



6i* 16 

<^o 54 
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59- ><> 

5 8.- 3 i 

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«i 

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950 

970 



^3. zi^ 5.(5 . 32 
34- '3 55 47',( 



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35- * ,54- 58 
35- 54 54- <* 



36. «2.53 8 

37. 58 j 2. 2 



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980 I39. \6 
990 |4o. 57 




50. 44 
49. 3 



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nincfupp^niturq^arts ipuusmuEs^ \ioc e^ fefDidiaai^-^ 

cer citcilUR^^pofitionis g.proiefionm-cfe-j.ooo.^"' lUKO^'^ 
rus fuppooioir edaqn pro fiui toto in TabuU ^uuafck 




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Qua^idoerso 
ma^rnoh 

quahapajrs,hoc:eft/?[rjcj4^ Eij^xit^^ 
behinu: quatftitsis arcus ^r^. ^i. -Cijius fti^iiris ^. \6t^&ix 
meQfuf a anguli cde^ Neir^e eleiiacto (]pae£ta > iuxia quam 
fietpropofita amplitude sb partiit» -^^^-^ol taUuin (piaui^ 
maxima proie^o intcgrs^itt 4000« 

ferfoidsfii^u^idiMldsf^ihl.^ \''[y . 

Datf ftmirae Ataplitudittis quartain/partem iutne \ h^^in - 
tabula finuun>^uaare ^ arigunq. ipfi i^e^^on^^ntein biBtrramfb^ 



ca • Sic habebis eleuationeqif qua? 
cit .$c4 fiifius baacin iSBquentib 




^onqusficaQi&^. 



■» » 



.r- 






\^ • 






^tf 






f^ .. v^ 



yiadclk pnBceifeate Tauola » 



jgedflfifi^fimu deUdpiiudtd dd vms cMrmd JSd pe^^ 
eftfhfi^4^$i.fdlfi geomeiriei. Vegliefiire com U mede/md 
vn4ir^di mMmers^Mle; chtriefcafer dfpume lemgepdffk 2j^0. 
figUoUqudrtdfMtedi li^eSdqttdUespo.dgtidr^kftiUid^ 
meldj e vede dirimfette ddeffe mmmere U elemdxieme ddddrJHt. 
dettopezM ejfirgrddkig.emmmiis.omerogrddirr. em^tmti 
ysfmoum^lememto.E dicoftrlecoredimoardteytheilfmdet^ 
fofetjaocmnmddiiltie^emt£ elemdxdomi tmrerdUfdUd loMd^ 
mo^fdff$^ 2jfo .foprd Vorizomte . Se beme qmeUe elemd^iomi , U 
efHdCfdffdmoilfeS&fmmio delUfqmddrd^ momfifomgomofer 1'm 
iigUerie^ mm foldmemte p^l^vfo de* morsdri^ o irdkoccAi^ ofdL 
fdm$drtimi . Demepork dmmerihtfiche com qmeUd frimm eUmdSUo 
moUfMdfara vmdfirddd b^d^ $md veloccy 
eome U Umedfegmdtd a » e com^itr^eio grdmde 3^ 

erizomtde offortmmo fer sfimddre mmrdgUe^ ,/^ 
ddrt dhro in^ulfo Uterdle ♦ Md com 1'dltrd ^ 
eleudziomefdra Ufirddd b« Uqudlefdrd fi- 
grd dim^to ori^mtdle ^ntdCom^diimtpeioferfemdicoUre mel 
Jfme , ofmumfer sfomddte^oUe ^tttii , </Ser dltrefdffdte fef^ 
fetuUcoUri dU^orizomte ; ouero fergettdrrobbe in vn certo de^ 
termtimdiofegmo , comte farebbero fdcchetti irmbdlldii com cerdet 
fiemi dix>ol/o , o^JdUitro , ofdrind ; omerofdUe com lettere , (^ di^ 
tro demtro . InfommdFvnd , efdUrd eUmdziome , che egmdhoe 
tefiddiadnte ddl fefiofmnioforteri UfdUdmeUoffeffoUogm^^ 
fero com Ufrimd , emimof eUudS^ne cdderiim terrd (cormeef^ 
fidicomo )Di ftrifdoi e cqm Ufecomdd e mdggicfe' eUmdl{iome bdi 
tefk qudfi ferfemdicoUre . 

So che rdrijfime voUe^ efirfi dmco rmdis^imcomtrer} che ilrmdf 

firmotirod^fmdriigUerUfidPerdffmmtoqme^fd^^ooo. comte 

fdrchefifuffOtfgm melcdUolo deUd TduoUmfird, t^dmcoim 

^tteUedelSigAGdUUo.ferltUdefidTdmUfotrebbefdarerei^ 

:• *E c iiU^ 






\ 



/ iU . Md noi mofirtretM che tln«mefo(itffoBo dii-oo» .ftr cA 
non/i rtie ad 4U«irSm'Ju laiiafmn ^hftiiuiiifopfi rmr a tmt 
te vnintrfalmente . Bifogna dnn^ue annertne ihe qnel nttme' 
tpfnf^ofh 4U if ^0^4 iiPi» l d*ftfgi f ntf diuuimh «fciircM , nt di 
MkrAdftermiMttr mfurr ;rtut.p ktiti difmiM^i^itdi qkdU 
iffef^an4>i\htfet^f0tind0cointtmrfiMtttt!M^ k-fnfi d^ttii^ 
fitr*ftfjibiii,fMnno la TanoUif^fitrt^tdntd^ftj^d^cUtthitiei 
^tfMt9ftrintmi>m/)ia&tJiri.,£ftrditrtitm^f<!mfitytme ei- 

UtftfiffkdddMUfe 4 tmtt le fftxJedeitMti^trie , tridttrre 
ltfmiMnratitin.faff%emttru^,fkemMtsi,>,.^ j.- 
- ilm^tm tirt di-vn c anvtnefer effmtmcitkfdttn-ti»tte cbe 
Htr eftr^pi^»39v,eVoili«to»Ufi*fofarevti4ir»iifm4- 
Ufttdifap-ttio faccit cott . St il mjtj^M tir)i. t^v o.mid4 
t^o . itroqntfit» > iinttmert zeto. ma^medeU4TttitUe4h<tm 
di^} ^faccio toftragiiotte i eir^n» 37^^. ilfmdnttmer» cercntt 
ntUttrMoUfirkrottAfrtt.s7 0. e sSo.Per^tutofrMti» Mgtmdi" 
ziotaparttfrofor^nntile trtnera^ farco dtldafna eteitaxitme 
doner efere^adt r t :in cire<4i'V0ero 79.fn» etmfUmtnie . E 
cosi e certo ebe quetta tale arti^ieriai ta t^ttde elenata d' a.fmm 
ti tiroMafaffi 2300. etenatagradi / 1 .«mero 7^. ttet fnadrtiiite 
ttrerafa^tsio.comtdefiderantnho^ - 
' • pitvJ^efoiitdaUitnofOtreibefaretfdi^citeiitrwarfii^affe 
iMtiSJtii firo maj[tmo dett Kttii^Heriiis mtfite 



^me da qnatnnqite tirofatto antoiafmalmeme^fiftffm 

ironHr' ittiro^ma^mo ditfnfe^a^artigUeris* 

'•■.."' ■- > ....■- 

^la vA pfiko dirizzat<) conforrafe } 
T^k Hnca^fjdellai^lefial^ielcuii' * 
ilerie l*aiTgoIo *4r,qtialliiique fifiai.^ P 
Si mifuri detto arigolo ron la fqua-^ ji 
ft^ , « tfouifi f)er tfcrijpio gr^ 5 o. poi fi 
f^Yt l^Srti^eria, fe vada ik palk fino nlpOflro * , eii 
lf|eirtteiri«nife1alittca *>^ chcfiapek'«feB[ij)kJ,a4oo-|itfffiC 
idetriti .< Dicotiiedatequeftecliib^e^yci^lacl^aiiien 

> \ laluii' 





4 

ff la Itinghezza dcl 6rojaaJiMc\lg4^Mt/^ttd9tla data anco U 

■ mvftnconeiiMtW) Omii^if»id^ Pfopoittioitr 9« dcpro 

A : E^mdhdato l^i^igolo della eleuazione c^t^ff.^o. iari 

I ito\aiigoloi3ettangolo £ dcds/)^ in fpezie .* e perche e dara ia 

I ^j^Jq paift»^jMUidaf«ia^g>iq«im|Mirt^ cio^ tfoo.paf- 

(i • Operefemo dunque cosi per trouar ia quantita di 4 d^ pcr 

via di calcolo, e de i feni* 

i^^KciaiiV Comeil^norefio 8^5oa« deli^aogolo d€€ g£^ 
6o;^oa0i&dd[Xupplenient9 idelkeiiHiazione% aiiaco dt^ die e 
6oo. Cosiil^Oyptsak x2odj«««>^iraquano J)iic^ro 695. 
£ cosi la hipotentua d c fara pafli 69 3 • Ma per che anco il tria- 
^tixiaist^^ d»iOlK»lQ;]qo^ 

i^Qgolo dato ririhi fijyuayjQflg A4f, allarettj^ 4ffi^ ^ trouo 
tf93.paffi,c6$iilim»MtalcadraquartPOwnei£Q ^3$^tf.4cco 
silarettacettatajtfr^itfafapffi ^g^i^vMaM^ ^i/.eflrndol» 
.lineadelPia9cfix>:^UH^^ raaiEmo 

tiio, fenoi radd(^^ceBk>Jt3:8^yen:a<a i^fi il niiQiero.di 
a772.paffirchrtanfia%4^^unghezzajckimafimotiro» cfae 
£cercauadiquellamachina>Iaqttale eleuam gr* ^o» fi trou^ 
tirarpaiIi24oo. ^ 

Maconmoltomaggiorbreuita, econvncalcolo folo po« 
tttna^eraretofiMP^g^6€hi^ iifcno totalct^i» ^/> f^iwi* 
u;kf4^&cfdk.m^emvj^i^ 

€raddfuofupplemento# Facciafidunque. Cpme il feno rq^ 
tale^al^la cfaith ^oo,cosi25094o.(cheelafo/nma d'anH 
tMdaeleiudette.Taoge0ti)advnquartoc\M ti^6. ^co^ 
latetfft.tf^iitcoueracomeprima xi$5.p9^.;laqualeraddo|^ 
piata daracO]|ie ibpra la miiura del tiro leaiirjstco > o ntjijSliaio t 
coixiffji^ogliamchiamarlo. . 
PtrCirMdm fifM MuertirechtqHtHo }ilmd$ ^srgomgm , 

tgre y dd qudfiMogUdtijtje d^v^drtigUerid^ qMS^to U meaefimd 
fidfer t'm4tite ditinst^fer UttedferfemUetddre ifbefdtk qudnte 
\ ' E c 2 IdU^ 



t20 

idlintd ^A.fitnMdfwndM}M^idia$i$ . - ^ ^* . ' >Mf^ 1 

LafitfiiUned^di ciimfegnadd^ifMMtksliietd^^ 
hclAfcidf cddnev^afdiUtTdniglmUyjdt^ 
CM il mcdifimo impeto^ che conftfifcc U Sfe^a arsigUehd AjhtJ^ 
endoperofiimffedaWimfedimeHio chefM^ afpenme Userfole^ 
zd deltdtid y cAefdffidmo effcffeHfiMe fer vdrfdte U ff^f^-^ 
uoni dimojhdcede* jJri^m A mo U ofimfeeM^dird^neSo ^fu^ 

&)iy^ iv;» /e/oie ednoie defenifo^umofdfefe U mdfimn jdten^ 

xai dUd fndle yfernennufer drid UfdlU di vm tiro* Ddtd 

ferotelend^ione^ olonghegp^dieffoiirsi 

Nfitt^fte(Iaprecedentef^^i€adaioranTOl0dpUa de^ 
uazione r4^, elalttnghezudeltii0 4i9*Sicercaral» 
cezza mafiinteiiallkqtiale ^peruenutalapaliaperada: Eqiiefta 
fara la linea ecVt^Q. pure di nuouo^ tfoapa(fi,cio^ ia quar 
ta piarte di luita la lunghezza 4 i . e poi fiuxisdS ; Comeil leno 
^tftfoivdeU^angol^ d€d fiqipieinento idcHaetcQaaibne, aUa 
#€" chee 600, pasfi>co5i jfocbo* iemMkjUa eleuazione ed€ 
«id vn quarto numero» e troueremo ^tf .paifiper mifura ddFal 
tezza c ^cioddeilamaggioredtczzaallaqualeiiapcruemita 
}apaliaj>eraria« 

' - ' Bene dnMertirfij ehe nOnfemfH fi ddofrM» le drtigUenitdi 
^dnierdtdle cheUfdld^vddddterfhimtre ntlmidefimo fidmo 
oHkontd/e , dd/ ^ttn/e era fortitd^ficomefn^ngono ie Tdtto^ 
ie delOdlileOi e nofire » J^iro donendofi tird9>tfifrd vndffUggid 
d^vnco/ie dec/ine y ottero dccUne \fdrimdttu donettdefi tif4tH 
^dlidfmttHtdd^vttd RoccdfnH^dttofotti^fio orixdftttdir yfito 
bofd non hdb/iidmofcitti^U^lonttditttOfmo dUdtmifitrd tU qttefii 
tiri . Potrebhe cdleo/drfild tdmoU > tttd cit^cnito /dccotget^d eite 
dotietteh(fi^neUdeottiferreferogt$igrndo d^elettdtsione delftS^ 
J^f efoifer ogniffrddod^incUndtMnedeiU/pidggid^efer ogtH 
j^fio ttdUez^ delU Roc4dy ilnt^tifU^o^djod^Ut qndfianin.^ 
• •* ^ - -i jinito. 




111 

'- Vnatale^glkmtonladireztioae dB. 6Ji 
VXQ 4c d^ foprail piano horizontaie dd^ Maio vo 
glio tirar e foprail piano d € incihiato » e cerco qua 

toiaralaiui^iiezza 4^ deldenotirofopraqQefto* 

piano^SidJr^4^pdrilpi]nta^9&i[/ a ; r 

co c . perpendicolari all'orizonte j e (icongiunga / ^ . k > 
/>.qualeperiecofeinoftface^paraUelaa}la diL ^ 

< Mifurifi con qualoheihumenco |!an^lo dd^^x cioe la ele» 
iiazione della fpiaegia,edallaTauo1adiDMeamplimdinificr(v 
ui la kmghekka deftiro ^oriaontale dd. Dopo quefto &cciafi. 
Coine V^ cangente dell'angolo deireleuazione dcll^Artiglie 
ria»alla ^r^cbew^la difoenzaddletangenti^ieduoiangoli 
ddby ddt noti , e(rendciryno la eleuaicione dell^ardgiieriai 
i^Saloro la eleuazione deUa ipiaggia fopra l^orizonte ; cosl dk 
noca in paflly ad vnquano ntoaero \ e ii trouera Ja reiti/d in 
p4fli • Faceia£*poi di nuoiio * Come il fenD.^0bIc,a qiiel ^tro^ 
liato qUartonumero chej^mifura di j^mjp$JE6. ycosi Ac &4 
dance deirangolo/^^r , ad vn quiirto , e cosi aueremo nbto il 
numero de^ipafii » i qiiali mifureranno h iinea d Cy cioe la koi- 
^czza del tiro che iara qutUa tale actiglieria fopra ilpiano^.^ 
j|i]ando£t]caairinstt^ : ^. 



4 . t t 



<■ .MaqoaftddxhdpimtOAbifogDairettfaregiuper 
vna^ia^iaidefcendenceoome ^b . Cosi trouere- 
mo la quantita di dby cioe doue rada a ferire la p^ 
la « 6ia data la direzzione ddy cioe, fia datol^ango 
l&deUaekuaEionedeipczzOy^^r, iiadatoanco^ ^ 
ral^ngoIo<lcilaipcliiul2ifone4^ l V%^ 

hniiKsqginiamooi^^orizonce^^ ; : . /i i 

djcdlariad%fele:f /,&;iftj^» e congi&Dgbiamo ci ^^XijuSst 
fara parallela alla dc. Hora fu la tauola<teUe amplttudini tMt 
ttereihb^juanti pa£&fia dd^m^ noi ceirhiamo quaQtt^fia dt^ 
!Rnaj&cciaiiil^cadCQl0»M Goifi&/4/;tangemi^ .d(^^ 




\ 



imzioneddpiaiio^cosi ii^notaiii|u£^»4difa^(MBKifita^ 
xp . EtauehcBlMkiJufnrft idt jtit. in j»iffiiieper6itfK:Q(Wia,'la^ 
jrr farinocaiapafli^ £accu&(^ Qomc Sf 

fenptocaleiaUa :^^no^)bfdffi^coiijiti^4^ 
r^^advxi^artxpiinqi^^i ei&ff&bmiAiraisr^ dell»^ iictta 
^i^ inpaifiiooeiajuc^b^^ta^iiol-^io^ 
dehte-^i^* . •. ., :»;Mi.;.M<>lii; .nhl«;:ji' •' . •: 

ditonif^M^pmneiiimmtgiifi.d^Gjif/^Ayp.i^ ^ f^l^ro. 

jPeri Mcoin qutjia <4fif^4mgW.fm§Jl<Mc0hlr^.tron4rV.M- 
ic^l^di^udfMnm40i€melfMdH^\mmQfnir . ' 

. Scaitf direaBioQe:iUi^zo.la^'ai^^ >. e.l?giiuep«v K^»^ 
ilniurodellajtorre^^ perpeiidieQiac^aU'Qrit£0itte \ t iia ladi'* 
ilanza ^d noca io pa/fi . Immagiataaiociii^heila pcdkNpafltli* 
ber^ ienzabattere nel nfiuro, e vada^si Iqqipine noU^o' ^ 

rtzoQteinr JaTAUoIadeikaffiplkialioidaiaiju^. ^ /\ 
utadelladi^. MaaoicercfaiaJiioiiiittC2^ai/^3i-* 

rifi ci penpendicolarc all^orizonteiyj&rii^paralle-* 

ia ad ^ ^> e poi congiungafiy^i, la qiiale paffei!a|>€r * A j> ^ 
la comqn iezz ione della parabola » e deimuro > co- 
mefipuoraccorredailecoregiamo^rate . Facciafi hQra>co^ 
me cd lucghezzadel tiro orizontale, alla c i/diffisnoaza. tsa||e 
linee dCy^dy^xz note , Cosi ia ^ r tangente deli^angolo del- 
ia eIeuazioQ.edeiraetiglicria;al&i<jrtangciite ddP^auDgokf/^ 
^. Itacdafidiiwouo. ;€i(HaeilXacK>.totale>tdIa 44 nbtaui 
p^sfi^ cosi )a gid n-ouata tagentedell*angolo/4^>ad vnquar* 
to numero i il qniale fara Ja cercacajnruTuca della retta dt iti paf 
fi^ etroueremoilpunoo <,fielquak|tfiderebbeaienrequel 
^o ; JLo ibafsofcalcolo fi pud.anfiQjoidunrcquandoitan^ de 
Bonfiaper{wt]idicolar<$» nu a fcarpa»*CQme quelliddBeteoder 
ffe ^Mtf^sftes ana<lubiaQdo di app^^ 
iciero Ja cura di cidaquel Geometra che fe ne curera • 
\ ZcMflUiulinideUefdrdkoUydellcqMMlin^uUG^^ 
jM^ccmCjpff^onM^c il^mQMniwnimJk^ifiMcdiVdcMm 

fdgnn^ 




I 



f 



« . 





/edA0.^ ^tH^ii mmfarJk^m^\ fe MnqmMmde fdatsigHendJi. 
menejp c$n letMeite ikvnjtfifsA , ficbe U imcM *oem^se ferdf^ 
fmmip mel iiMeUchdiUd cdmfjgmic U4ftrihecikM»ficiifiMmd^ 
efer che i mi vama a tamimare neltmzo»te\ che toccd / m)$u 
mAfmr4tatUUcrMt£^£erche¥em^€epme(iticameBtc)i^^ 
^tilmn^JivntmklsttMAiJO^c vigiimnditeorizM&ttt^fercji^ 
pne deU*dUexS{d dettd heccsdei fexAofofrdHpimmideMMcdm 
fdgnd\; Pdrctheilfemididmetrcdedterkoteyetdgr^z^del 
metdilo$:agioninocheldtoccddcffMiglieri€ ordindrie %rengdL 
diUdfofrdiij^oi^iziontdieintormToy 
dnehrdccid^ Snffongdfi dmnqmiio^. 
a Id boccdd^vndcotubrindy efiktho 
ritLonte hc.dltezzjt deUdboccdJid 
td rettd a hfuffofidjM hrdccut^ fjr t^ 
fdrdbold acd fidittiro tineMdto;fi 
cercdtd rettd hc^ Sid&titofkmi^ 
rettOy omdffimo dtUdmedefhmt nrti' 
gtiiridytdfdrdhotd acf> efongdfi 
ohe a (fidsovo pii^Gewmetrjiki^ iioe tjooo'. 
tine\. FdccidfiitJhnicirsotofo&todekdProfdfiiione f\ ii b K (^ 
froihttd-^i rgndledtid ziy fidf^ticbi Idc^ Rertevoji mfirjth^ 
HfdrMnttf^ a i imfetwdeBd ptrdbotd a ef^ «Km^.nAMb a(: d 
(fmh^fkmdei^anidejiinj&ndchinmjifert^ai 1 fdtd jUjfMrM 
fdrtedttJJooye^doi(dfdrdiiim^cd*^nf^ idfitrmid^, 
fid dilai ntd dncht a ifimofirhpkdoffi^d, i > ferpfono r- 

S^/fi^,^ v^f'y^%fengonfMdefierdldietreJimd€:t.tmfl^^^ 
-i(/ii»V<^ai> 2^ DnntfttefhJiotfuxmofertd^fgoiddAt^^ 
f^mtimoi^fm^'^<^h^ tMe^tOffi&i^mmdrmt^did^^ 
dd wdttro nn. troneremo do o o o il qudhfdrkdtfmMJtlkifHk 
tdhc;e cdttdtdne td rddicc^sfitmdttdi UOBOtrem^ chmdd retttt b c 
fdt^Jt4f.bhohim: Cdmcltk^fidmtifntcke tptoUmmdc^tmd >^td 
e^tmkfi^nUffm^ tiroJf ijoio ^. brMCMiJkimirrJ idboomiJH;. 
ieuditmdmm^ irrdiitUfifediMmjtltte ^fimtS^^ 
mm mimtt jd^oitttmi^ot^^ ,imtgo limsoigm wMr^fSMnU^. »4^ . 

Aw .74 ^dm^ 




UymdinclmMt0Mfm^t9i€r9J»^iM^ftrtMgi^0md^ deBt 

fUQttit ^vnbdSitBe^t d^VHd Rtccdy tdiaiuUmqutdtirtfit^^ 
^he IdfoUitM ftftd ilfidnt htniitntdiey fi cttchttd i» qtufit 
mtdtm '■ • ' ••• i* ' . . . ^, 

\. JEctrtt cht dtutmdtfi titdttddSdcimd ctvwd^tc^d i tmtrt 
di w cdBtUtftfio in cimd d^Htnfkfit , t dd qmdlm^qMt ImtftJ^ 
ttfmlfidm ttizontdUdeUd cmnfdgtidfttttfo^d^ ititi timfri^ 
tdnnodfidifiulmngkicheinttdtifoftdldtdmoUdeAt dmfUttt^ 
dini te qmefid difitttntdfdt} mdggMre tdmtfik qmdntt fim dlm 
tdfdti Idfitmdziont dettdtti^tietidftftd qnelfimnt ttiuktdt 
ntlqmdU dtmonofetirlt fjdkj€tetmindttititi\ 

Sia l^^lte^za della Rocca»o d^al 
tro luogo gf. e debbaQ dal poma 
/.tirare fopra ilpiano dellacam- 
pagna^r. Iminaginiamod 1*0«- 
rizonte/^, e fiitto il tiro/(f ^ rcon 
qoalunqueelcuazione» Kicercala .. 

ipifuradiff. 

DaUaTauoIadelkamplkudini^troiuerdtA <piantita di 4h 
e daila Tauola delle aitezze ii trouera la ^ ^ • altezza della pa- 
rabola» Lapraticapoidelcalcoio fipotrafare inpii^ntodi* 

^lmddtetmt nmmetms a b ; mtddtdtmfq;jdimiddtmftt a c » <f 
qmttmsttitldtmsteHmmfditMtU fcb • Dtttdtm ddndtqttm* 
tttsidmdi&ttSyin z^^&fttdttliitddixqMddtdtd ddbit de* 

Ouero potr emo operare cosi • 

J>mcdntm fimmlnmmtti d c, c a , &ftodmiiitddix qttddtd^ 
tdttittttttUtUctfttftttitttdUsittttt dc^cau Pidtvtcz^tut 
ftddi^mntddictm^ ifd ab dddlimm nmmttmml. ti^qndttmswn 
ttttttts ttit itttttm de. 

Ouero fioalmetite a quefto modo ) 
' ridtvtnnmttMs cz.dltitndofdtdkthtttttAnUt. Adtttt^ 
mtttmt cd dititMd.fdtdhtUi^dttisfimtdj Itdzh.ftnMtt^ 
fHtMtUfdtdhtUtxTdhMUy AddUtpn qMdtttnn nnmttnm^^ . 

SMntMtttdtindtntmttfMstntdit Uct fttfttrtifttdUsJtottrdi^ 



f00 t m MU M rms &i»tit ftb> md4midiitsilUffdfMi$m^ 
UsixAiie^itip/kmdc. Ei tmn d g (qHdlis ^fi^b mHdfig^ 

Ma potrebbe fog^iuiigere alcuao che dalla fommica j^il 
4>eiro A>r(i occorrera tirare con KardgUeria inchinata all^ingiik 
dieali^u»per6 farebbe neceflario^fapere per regola Geo- 
inetricalaluQghezza de'tiri» iiche (iauerd inquefto modo» 
Sia il tiro da farfi aii'ingiu il fegnato fk con aualunque ango* 

10 d^inciioauoae fotto l'orizonte , fi cerca g i . fingafi coni'im 
maginarionecheiltiro abia da farfifopra lV>rizonte conla^ 
medefimainclinazioneperrappunto, &per le regole prece- 
dend fi o-oui la quantitd di ^ r come fopra, ouero di JA ,idaIU 
quaiefeleueremolagianota/W» ouero Jg rimarri nota la.^* 
quandtacercatag^. 

Ma fe data la eleuazione gr. 40* del ^.Jr^ 

dro^^r^ elabafe^r itfoo.pafiinoi y^ pv 

voleiitmo fapere tutte le diuerf(p altez- ^ — ^ 4 c 
ze del tranfito della palla fopra (^pialua 
que puQtodella linea 4 c . Faremo cosl • Diuifa per mezzo la 
4 ^ 9 & alzata dh . queila fara Taltezza fiq>rema » e fi trouerd fu 
la Tauola deile altezze , e delle amplitudini» operando in que- 
ftomodo.Nellatauoladelleamp&tudini dirimpetto alligra* 
di^o.dieleuazionetrouo la linea Mddfcn pani 9848» nuu 
nella Tauola delle altezze trouo la linea t d eflfere parti 4132« 
Poiperlaregoladeltre^dico. Se^ii/pS^S^midipaifi 8oo» 
conformeallafuppofizione; dt che^ard ^ija^quandpai^ 
£ldara?eritrouochelaretta^^dpaffi 33^« Siaprapropo^ 
fioqualunqueponto e fopradicuifivuole faperraltezza del 
tranfitodeIlaiKdla»cio^laIinea ih. Suppongafi che la retta 
i«r fia looo^&la ic 5oo.efacciafidinuouolaregoladeltre 
in quefto modo »Se il numero auadrato di 44/,che h 540000. 
Middilnumerorettangolodelierette di^icdit ^tfooooo, 

11 numero Mchefiitrouato 33 tf.chemidard^Eria:ouo i X 5par 
fiadunqueraltezzadelhparabolafoprailpuntorfupaffi 31;« 
Ghe ^uello cbc fi cercaua • 

Ff 34* 



Safteri tduer^^dctepmdt^ ^tiejhfetdfir il cdceU M dlcmm 
%Htrietalequ4Hpoffon0$ec0rrereinfefne4queHitifi* Petemd^ 
noforfialtri cdfifimili d que^i^ e fdfticoldrmente i conuerfil^^ | 
\ rojma dalla intelligenzA di queltifi pofiino/dcilmente dedurre ' 

^ quelli y e l*ingegno di qualunque Geo metrd dppUcdudoui troue^ 

ri minor dijjkoltdneUofciorre molti di quefiifrohlemi ddferme 
defimo , che nelfdjfdre te lunghezd^e , (^le ofcuriti delU noBrt 
efflicdzdohi . Peropafi^eremo allafabbricd deUafquadrdy Id qmd^ 
lefare veramente affrofriata , anz»ifatt4 dalla naturd d foBd 
fer mifurdffcientificdmente , ^ Geometricamente i tiri defto^ 
ietti . 

J)ELLA S^ADRA. 



Rlducafi ora in pratica , e fciolgafi per mezzo di vno ftru« 
mento alcune delle jgia dimoftrate propofizioni. Fabri* 
chcremo vna fquadra militare , la quale con certezza inuaria- 
biie infegrii (almeno alli Fiiofofi Geomciri,fe non a* JBombar- 
dieri pratici;) quanta eleuazione debba darfi a qudfiuoglia.» 
machina » accio ia iunghezzadel tiro riefca dellaproppAa mi- 
fora . Sciocremo anco per mezzo d'efla lutti i Pr oblemi , cho 
fopra il tirar delle arti^lierie fipolfino formare,- quali gia ftro- 
n6 promefli dal Tartaglia , e poi ridotti in Tauole dal Galileo, 
c6 alcun^Ultro dipiu.Si accorfe l'induftria militare,che IVfo di 
vna machina tanto nobile , e di tanta confeguenza , quanto h 
rartiglieria , farebbc ftato troppo riftreito , & di poco bcnefi- 
cio> fe queVa .no fi fufle potuta adoperare fe non dentro a qucl* 
li poca diftanza^ ch^elia tira di punto in bianco^o vogliam di- 
tedi num jfenzadargli con la fquadra aiuto vanta^giofo di al«^ 
cuiia eleuazione • Fij pero penfato come fi piOtefle lire , accio 
con quel medefimo pezzo, il quale per fe fteffoflon tiraua piii 
chea.cb.ouerb^^o.pafliGeometricifipoteflredramt e 400. 
i&ancoi ^oo. tfih. ,c pii!kyfinoallaiun^ezza del maflimo tiro, 
diepofla farfida quel taIepezzo#L'inuenEionefii queftfi \Qt^ 



i«7 

iiiificiiu»iK>ailaiutareil|Mtt^ nonlo 

diriszauano a dirittura ^txfo l'ogg€ltei iachedoueua colpire $ 
ma ceneiidolo nello fteffo verticale doli^oggpttOj io eleuauano 
fopraquellalineareaaj laquale v^dalpezzo all^bgsettoic 
cio faceuano ora piu» Sc ora meno » conforoie die la swnatur 
n del dro doueua eflere maggiorei o minore « Ard^o che fi* 
no dal principio del mondo t ftato noto anco a i puttiinefper* 
11 • Vediamoche douendo effi con vna palla di neuey o aal« 
^ croi colpire in vn fegno viciniffimo, la fcagliano a diritcura ver 
fo eflb fegnoi tna douendo poi giuocare a chi tira piu lontano^ 
ouerofareafaffitra diloro^ nontirano gia orizontafmence» 
ne a dirittura verfo i loro contrarij^ma voigendo i colpi a mez« 
2'aria>fenzaattBr£ittoalerafpeculazione,tiranotutti all^ele- 
uazione del qui&to $ & afiCQ del fefto punto della fquadra mi« 
ittare aioro ignota* I Sombardicri poi ebbero col progreilb 
del tempo vno ftrumento» iiquale faoimence miiiira quefte ele 
uazioni. 

FuinuentatadaHiccolo Tarcaglia Brefciano Macemacko 
infignevnafqoadra con legambe difi^uali congiuncaconil 

Suadrance^la quaie gia pi^ di cento anni d fempre ilata in vfo i 
: e ancora IVmca regolatrice de'Bombardieri , non folo per 
adoprarPaniglieriaf&aizarlainquei tiriy che eifichiamano 
di volata» raa anco perli uellarla negli orizoncali* Diuifeil 
Tartagliaquelquadrantt in la. pardeguali, cominciandola 
numerazione di efle dalla gamba minore i fuddiuiieanco cia- 
fcuna di eife in altre x 2«)>arti eguali » nominando qoelleprime 
Punti » e quefte feconde Minuti della iquadra • ponghiamo U 
figura d ella fquadra, e moftriamo come ei$a mifuri meuazio^ 
nedelpezzo* 

SiMl'dni$m4 delC49$n$n€ z,h.fcrm» 
in qndlche fofitmd ; Mettdfi in boccd 
Jteffold mdggior gmntd delldfqnddTd 
ca .fichefidddd$tifn*lfonddmento di 
dettddnimdy e cdfchiilpitmboind Jo 
dicocbeCdngolo tQd^ciofil^mco ed, ^ IdmifkrddeUd elend^ 

Ff a 7;^ono 




I 



%i§medelpe7^\ Tmfi vmd mzemtdle z£. fstMm gfdmgeR 
imefMiUfunto ^^retu^mdMmcol^dmgele zcf.iremyMdmM^ 
qme gtdngeU c a ii& iC^fono eg$udi ftrts. delfefio . Ontrw 
eosi . Tirififer c torkLomtdle hi. Se ddgU dngoU retti h c c^ 
a c e » filemeriil comune a c d » refierk VdngoU e c d deUdfqtt^ 
drdegndU M*dngol$ deW drtigUeridfotto l'§rixMte h i » ofofrd 
feriMnte a f , che e lo Bejfofer efiire dlterni . 

Col mezzo poi di queft a fquadra fi e fatta dalli Bombardie^ 
ri con lunghe olTeruazioni vna pratica tale» ch^efli fanno ouan* 
d punti debba eleuarfi verbigrazia vna Colubrina da 40. 
per colpire in vn fegno ioatano per efempio p^ yoo^geomc- 
trici , o in qualunque altra diftanza • 

Mavagliailvero> le ofTeruazicHii fonotaiuo £dlad; fono 
cosipochi i Bombardieri che le abbianoiatte>eleabbiano 
iatte efquifitamente» che IVfodell^aitiglieria > leuatone il tiro 
di punto tn bianco , non pu6 auere fe non pochiifimo di certez^ 
za • Volendofi acquiftare qualche fcienza ficura intomo alla 
%iadra ordinarla , larebbe neceiarto di hxt l^fperieoze non 
folo contuttelefortidipalle» econ tutte ledifferenzedipol^ 
uere , ma in tutte ie fpczie de i pezzi » & anco in tutti quelii^che 
etfcndodella medefima fpezie,fono diiferenti di grandezza» e 
poi a tutdi gradt delle eleuazioni pof&bili • Molt^lico > che 
quafi anderebbe in infinito > E notiamo , che conuerrebbe fa* 
tc qu efte efperienze tutte ad vna ad vna ; poiche no e vero che 
per via di proporzioni fi poCfa da tre> o quattro tiri di vn Can» 
none ^ &tti a diuerfa eleuazione 9 argomentare alcun^altro » ne 
pur dello fteflfQ C^nnone caricatocon la fteisa poluere>epaila • 
Ghequeftafiacosi, fidinioftra per mezzodelle Tauolepo* 
fledalSig.Galileo>edanoi. Perefempio. QuelCannone 
cheeleuatoalfeftopunt6tirapafli4ooo.eleuato advn punto 
douerebbetirarlafeftaparte, &a duepunti la terza» &atrc 
puntilameti. Malacofapaflfamokodiuerfamente. Perche 
eleuato ad va pwitDftira lo^x. incambio di 6tf($xhe ^lafefta. 
paftedelfudettomaflimo^tfro 4000» Al fecondo ^ioto poi 
(& ^ifl^fi xhe coB quefta ^deuazione 1* aitigtierie m-. 

nofim- 



ao fen^ k meta dei ouffimo tko)cieI cafo 

incambiodii333«che^later£aparte. Al terzo punto tire- 

ri 2824* in cambio di 2000. che ^lametddeimaffimo tiro 

Alquartopuntotirera 34tf4.incambiodi2tftftf«Alquiiitoti* 

rerajS^.incambio di2333,che fono cinque fefti dt quei 

maflimo. Vedefi dunque come accrefcendo eguahnent^ le 

cleuazionidel pezzo > cio^ tirando prima ad vn puneo folo,poi 

a due f6c9.trc^€ cpiatoro &c* finoal fefto, gPaccrefcimenti del« 

lalunghezzadeithi non crefcono eguaimente $ cioe con la 

medenma proporzione con laqualecrefconolceleuazioni/ 

Ma mentre il primo punto tira 1 oj 2« il fecondo accreire fopra 

cfso » 96S. U terxo accrefce 824 ; il quarto 540. U quinto 3^dL 

Ufefto 140. Percauardunquequalche regoia dalle efperien- 

^c 9 era necefsarioil fade efattamente y a tutti i gradi della ele« 

iia;done» intutte lefortideipe;e£i, contuttek varktddeilc 

polueri, e le diuerfe materie delle palle , e fbrfi anco direi chc 

cra necefsario che le&cefseogniBombafdiereda/efte&a# 

Cofe quafi impofllbili a ridurfi fotto regole > e cauame cene;i^ 

zsL alcuna» fe la Teorica» e ja Geometria non ce ne daua mani* . ^ 

fefta fcienxa mediante queirvnica propofi^ione del Galileo » 

nella qualeprimodi tutti egli ha auuertito^ & infegnato anoi^ . 

chei proietticamminano lutti pcr vna linea parabolica. Su 

queftafuppofizione fonderemolo ftrumentoproniefso:fe poi\. 

per llmpedimento delmezzo le parabole ven^ino uoppo^ 

defbrmate , o permolti aln-i accidenti i dri riefcono inGoftan*' 

tisfimi , ci bafiera auerfodisfatto indubitatamente aHaiciiola. 

de Matematici , fe non a quella de Bombardierr • 

Noi auanti di porre la £ibbrica della noftra fquadra > qualc 
fion cofifte in akro che nel defcriuere vn fok> femicircolo » ik-- 
uideremolafquadraordinaria in punti difuguaii ^ di maniera 
tale che mifurino non le eleuazioni del pczzo , ma le hinghez- 
ZC de i tir j^ che h quello di che IVfo noilro hk bi fogno . Cosi 
aucremo cenezza che l^glieria ,-fe fara aizata ad v»punto 
dieisafaUadra, drcra alla lunghezza dVn' tale ^iazio , qua* 
limqttttfiaialzatapoiaduepuntirq^doppicra preciiamente 

quei 



l|e 

dro;feattepimti,dreritredlqiie£^tt4»rea<|uat^ s mez^ 
zottireraquattro etnezzo:feacin(|ue&vn quartOitireracm' 
que & vn quarto i ecosi fino aliefto punto creiceranno fem- 
IM:enelloAefsomodo>econla ftefsa proporzione ipunddel 
lafquadra nello flrumento» e glifpazi; dei dd nella campa* 
;na> e dal fefto fino al duodecimo punco anderanno nelia ft ef 
manieradecrefcendo* Lacoftruzionef e dimoftrazione c 
Geometrica> cauandoii dalla propofiziooe da noi pofta al nu- 
mero XL de proietti > la quale dalla data amplitudine infegna 
trouarPeleuazione . £ ferue comunemente per quaifiuoglia 
forte d^ardglieria^ e di mortarii per quaiunque fpezie di palla p 
odipokiere. 

Si4B0 U gMmte 
dtllafqUddtA zh Id 
n^aggiore^ (^ acld 
minere :foi/aiU cc 

tro in a faccidfi il 
qMddrdnfe cdc.fo 
frd il qudlefi hdnM 
d notdte i fdmi di'- 
fugudlii& intorM dl 
didmetro ac .fdocid 
fiilfemicircolo a f c 
/ tirdtd Id i^ferfe^ 
dicoldteddzbyetd 
geute dlftmicitcolo 
diuiddfi ag iufei 
fdrti vgudliferdue 
re i fei funti deUd 
fqUddrd^efoi cidfcu 
ndfdrtein u.fer 
duere i minuti (qui 
dofero Id grdnde^ 

l^d deUofiromentof^^ cdfdce di qtiefidfecondd diuifione.) Be^ 
tdfidvnddetUfeifdttiU gh. AtxifiU limi .fdrdUeUn gfc 

U qtu^ 




331 

U q^dlefegki iifefmvrtd§neifumi niy i . Thijifn ddcen- 
tro a Id rett4diid^(^Ufu»tQdf4riilfeBedeUMfqHjubrd. Ti^ 
wifiU ai U&ilpHnt9 i/^ftkHqmnte delUfquddrd; tkrifiU a 
m Q> cf ilfnntQ n ./^i ilfittimo , r w/ 4^/ tuttigU dltri • ^/v^ 
uertdfifm ehd^ofenClUenefftrkfikgiufidyfe dofo dner trond^ 
to ifunti r-^^3* <^f . fmneremo eon Id trdffortdzione def^ il 
nono f decimop & vndedmo . I mezzifunti > kqndrti , & i mi^ 
nuti fi trouerdnno nellofiejfo modOyCoidiuidere in me^o , oin 
fudttro fdrti, onero in dodici cidffund deUeforzioni deUd Uned 
a g . con dl&dr Uferfendieoldti ddfU funti delle diuifioni iy^- 
gnerdnnQ^detteferfendicoUriilfemifircoUy&ffrifunti deU 
le fe^onifi tirerjtnno iftmididmetri nel quddtdnte , cbe qnefii 
fegherdnno ilquddttAntene ifunti defiderdti^ de* me^funti^ 
de* qudrti , o minuti , 

Hora e manifeilo per U Propofizione IX« !no(lra)£befe 
ia linea della direzione » o vogliam dire della eleuazione del 
pezzo fara n o , ouero df ; ia amplitudine e lunghezza del jciro 
jara comc la^uadrupla difo j e fe la direzionelara d m, ouero 
^n. il tirofaracomeiaquadrupladi rm: e^juando laeieua- 
zionefuflefecondolalinea dfd, il tiro fard come la quadru- 
pladi qf. Malelinee/^,rip,^/. per la coftruzionenoftra 
egualmente fi cccedono , e p6r6 anco le loro quadrupte , oac*» 
jro i tiri fopradetti egualmeme ii eccederanno iVn 1'aUrOii 

Vfo deUdfredettd diuifioneyfdttdneUdfquddtdordindrU. 

Slaci propofia qualunque artiglieria > o mortaro^ e con eflfa 
facciafi vna fola efperienza i cioe fia eleuata a qicilunque 
punto , come per efempio al quinto . Sparifi, efiirtirutxlalun^ 
ghezza del tiro, e trouifi, verbigrazia , eflere 2 ooo.pani ^ fciw> 
^jueftopofliamo fapere quamotirera lamedefima arViglieria 
caricata nelio fteflb modo , & cleuata a qualfiuoglia punto^ o 
minuto , che fara facile pcriaregola del tre , eflendo in quefto 
ftrumemo tanto i punti quanto la lunghezza de i ti n prx)potzid 
mh. JdipraiicaigudUt Vojgllioiapere^<^ 



V- 



punto. F6 cosi ; fe 5 . punti cKedero s 060 paflS quanto daraii- 
no tf • punti ? e trouo 2400. paifi • Dico dunque che quella ar- 
tiglicria al fefio punto^cio^ col maffimo tiro > tirera due milsL c i 
quattrocento di (pielle pard delle quali al 5« pumo ne tiraua ' 
l^boo. 

Auuertaiiperocheincambio di £irequeftaopenizione c6 
i punti 7« S. 9* 1 0« z I . & 1 2« fi£i con i loro complementi, i qua 
lifono 5.4.3.2. i.&o. 

Ma fe ci fiifle comandato (Sc importa molto piu ) che noi c- 
leuaifimo il fudetto pez2o in tal modo > che la lunghezza del tt* 
ro douefse riufcijre per efempio pafli 1 3 6o.opereremo cosi* Se 
2 ooo. pafli furoQo htti da $ .punti» o p^r dir meglio da 1^0 .mi- 
nuti di fquadra »1300. paffi da quanti minuti fi ^ranno f ecco 
Toperazione 2 ooo* tfo.ijoo./j^. E troueremo che per fii- 
re ii tiro di lunghczza di paifi 1300. bifognerebbe dare 
all'artiglieria Peleuazione di minuti irentanoue di Iquadra» 
oueio dipunti tre & vn quarto » 

» 

MA fenoivoIdStnoformaBevnoftriimentOfilqualenS 
rolomifurafseUIunghezzadeidrifani adiuerfe ele* 
uazioni, ma anco l*altezza delia parabola % la durazi<me, o tem 
po del viaggio , la fublimitS , e raltre cofe dtmoftrate nd pre- 
cedence lihro de proietti ; nitto fi iaxt col folo» e femplice femi^ 
circolodellapropofizione ^.Ma venghiamo aliacoftruzione. 
Preadafi 




lalamfflarec S 
tangola dh 
€d, di otto* 
ne, od'aIaa 
fodamateria 
laqualehab 
bia la gam- 

ba^f luDgadapoierttenafiiBbocaddpczzo. Facciafi 

^il 



*5f 

(^^sQpqpf,i^9.(k^kM*^m^ pofig«fi ii£locoii 
tlt>k>nbcl>cdiiu«fiUfciBtcirco(lo. sfi iii9o<ptftiegiKdUkte 
Cnanooli^o.^adidel^uadnuMi^Ouecoffl I44.f»aniegiul4 
cfie rarannoipuad, emoHKi ^gMali della fquadkaordiiiarii* 
MoftriamooraGeomecricatnftecomequefta rquadrafiaatti 
« miOiraiie iK>n fomnia fimpUdci le Iw^hezae k ralieEze de i tt> 
fi) Utea^deUeduraziotui ie Iut>Umiti delle parahole»4K 
icdieiuUuonide*pea«i .Epoiponemoladiuifione dellelinee 
ificr«fufema.auerWogoodiTaiiokalci»a peropenr dctit 
iquadra. 

9 ('^^(fem^f^k afb 4t 
t,tmffim*rknf^ttfktif4t, \ \^ ■ 

^4fei»t e 2Lf9yf4t9fint3i^f * 
te. imfer^ ehe hdMemde n^i. 
fiffU MniJleHe elelfemicir<. . 

(»U iep fe.f0tif et mi t t ^ •. 
rn^tUMe egm e(**egr4iifet. 4 
wefeiey Mi^^fyf^ht 



t .. »» 



i- « ♦ 




i »\. . Ji 



tm\^ifi4mifiie44tiltM.x\K . ; .n 2 

M^bai, ti9ed€iUekit4xi0»9idfieS^f^Mttmll{mni^ 
UeriKetitef^dfemfee-UkmsiU iHee elifikrehefet$9e/m> 
geremeeheUltmed iib. iismmMlfeimmeeU ^ timfe$9^ 

4eU4^efefi^mti^»em^eim9Mt»niddmdjamw9ri UU^ 
9*4 i\^^^fe»dkeUHilMdUmetn^fktkUtfMmu{fem^4kUm 

ioiefUi^fte^ t Ut^^^^UinfU b h.finkiieUimuififte^ 

m* deUdf^deUiU aii ^Ufrtimitii t Uhi,fmi4ltem»^ 
tt.edm^f^j^itie^^, .• -•. .-. . 

C fi Che 



^ 






>^ti^^ia&' \ 




gelU.f>0j^m^PtJ4Jirfigtiefm]^m^-ti»riM^ b/^ 

IMutM ntnfnu . md uaU d^ejpt tiro ie ccsi filtremifi^en^ 
femitmeU hi\pir4nnotHtieveret^r^m< 'ffifdMtififibtii 
tridngfilo h b f; eJtmiledltriMgolo yfm^fWie^oJmMiitrH 
tdnipU , epei4nerdne4ngoli4a4dift^h%^'Mhfti'm^lttme* 
depmefretorxionifkr4nno trMdikroimt^ ifff/hkyMotf, t 
jintedeUarqn4dr4 zc.neaetfn^Ufl^^mfitmimi^le^. 
%rt vtre neU'imm4gin4tOy ev4fofifaii^m-ii^t\^'^i^i H 
/inee a b , b f; fh , h.b ^tr4nnofr^diim itjrlffivi^i^ 
de/mceJkeA4nmfre/fettin4menteM!^}Wi:^}f»^Af»y 
an4ntodff4rgomen/4r neaefrotorxmijfVmi>iffitfiCM»tat 
trroreftrnirti.nAnmeno, deUe/ntefi^Mf^^di^ny ebeitUe ytrt 
imm4gin4ftneU:,4n^iexizddelf4ridK^]'''J:''^^ v 

Refta horaj che ponghiimo xoittcqaetta dottrlfia , che& 
qui h ftatamcra.4>caaajJfci*l^EQfliMi6Nir^!i jpratlctf itaatiQlii^ 
le . e con feci^td . Ciafche^i^ v«»fthtff>«raue!!noi coga^ 
lione dcUa quantitaWelle linee ^ hi^fip Jfi Mwo yro|*ftr 
Eioni nella precedente figura , farefcrb&tibcetfiif i6 , <Jhe tiftWle 
BM!4^ceJ«ficA'inflci^^ijUuiift*lDfa^ mikiiiMuncf tonfflialdkc 
€ad)«DrJKttfiir4««\^ qtwik) e0eKott>ei^il|atdei%mo di pHtf 
maa^ k «)(iyui0MOcit^iam«iBO idm^ S6 ii femidiaffletro e /, 
cSoii «pfnw«itMbdi%ataaK%ttab^k^^ 

flaMcginckl^ahdl^Ma^OitUiu^ ii^ in efiTfiporau 
]^reikn>^iid:«Vx^faM|iiB!beidfcd«Ilp liiiee\<&fc £ii»inho indi- 
ci dells lunghezza , & altezz» de iuk e ddinnto ddl^ngola 



I 



I 



*- 

^ 


•» » 


o:^ 


• 


» 




• 
^ . ■ 


• 






c- 


f • ■ '- 


-/: 


f - 


. i-^ 


* • > 


Ci 


4 # 



• ^ 




\4 

l 






i • . .'4 






236 . 

del remicircolo t metteremo ii ffllo col piombo « 

Quanto al numero delle particellet nelle quali (i doueri di- 
uidere ii diametro d k potra eflere in arbitrio dlciafcheduno i 
iaraper6 bene eleggereiinumero aooo*perchefitciliterairo- 
perazioni Aritmetiche • 

Deue ben notarii > c)ie fe akunp fabbricaflfe yna fquadia co 
mt^h detto , a pofia per vna fpezie d^artiglierie foia % auereb^ 
befenzavnaminimafaticadicalcplo lamifuradi tuttiitiridt 
etfa • Ladiuifione di quefia tale fquadra douerebbe hx& a 
pofteriori in queflo modo • Facciau i^efperienza del maffimo 
tiro di queUa raie ardglieria^ per la quale vogiiomo far la fqm* 
dra a pofla > e fi troui effere verbigrazia paffi|oco. Di&adafi 
poi il diametro della £quadra in patti 1500. « il femidiametro 
perpendicolare io parti 750. eguali ; cioe fingafi » e fupponga- 
ficheiidiamenro 4i^ i5oo.fiaiametadelmafIimotiro 3000» 
parimentecheilfemidiametro perpendicolare cdj^o. fiala 
quarta pane del medefimo tiro maflimo,e cosi data poi quaiii* 
que ahra eleuazione > fubito che mctteremo quefla fquadra ia 
Doccadelfuopezzo» immediatameote vedremo quanti paffi 
fara la iunghezza» e quanti l^altezza det tird , &ci Ma pero que« 
fla tale ^uadra fatta verbigrazia per vn Cannoofc dfa ^o» fareb 
be anco buona per ogn^alffo Cannone da 6b.che fbsfe deila 
medefimalunghezza» &aitreproporzionicome quelio. 

£ ben vero che volendo noi fare la fquadra vniUQrfale» che 
ferua indiiferentemente po: tutte le fpezie ,. e tuttie le^randez- 
ze dell^aniglierie > faremo cosi • Diuida^ neila £giira prcce* 
denteildiametto di inparti 20oo.egudi tra di loro» Pari- 
ftiente fi diuida il femidiametro ed.m parti 1 000» e^li fita di 
loro.(Noi per la piccoiezza della figura abbianft diuifo folo in 
1 oo.pigliado le pard a dieci a diecit;Fatto queffp fi tirino dalle 
diuifioni dellacirconferenza fegata ingradi ^|;ualialfoiit0t 
le emde parallde alU diametri ; acd6 fi pofl^ fc^ e0i diao^ 
^iieggere ia quandtadelle lineerette, confbmfeoceorreri* 

^ Si^ci ora propofta vn^artiglieria ignotay^ • Facciafi la pre- 
Mcfj^ncnzainqudllplDOdo^ PoBgafiioboccadieflala^ 

< Iqua. 




. <^iwdci 15 calcliiil fil« ip. tptluowiS 
Jupk^in >.f.tegj^^gef via .dciUuuLa 
guidalaquahdddi f» fulfcniidiaine- 
lr«<^uiJ'o>«/ituiga wMmxisi^cfoi 
ipuifi l'iirtiglieria, e A «uiiiri ii tiroiclie 
^aDo^^reiempio}!!;»., Caricbi- 
S <u niiauQ l'artiglicria pellofteflo n^r 
'dbici^iiialzidiuc^iuuente Mp^qcbcil . 
flocafchialtroueifl »». iicercil^luiigiieaKidiqueftoi i fO. 
' Faccialicosi. Seilnumcrodi it ddlalungheizadi i*j^ 
palE', ilnumerodi ml, chefileggeiul femtcUameliio diui&ik 
quaati padi dard? e troucraiparimeiiK Jallu^ietnileltira 
niimeratainpal£. ^ , ^, , : ..■■ 

ChIroleael'altezze,enon,leliMighcuffik'tini£Kd«It 
AiAa dpcraaiohc come s'c d.«Koi «aawin itop le linte J», ml 
chedannoic lunghezze, m^ fi bcne «bn :l&i<, ii, lequaU 
danno lc altezze . Sepot volcJSmo lciubiimita , biibgncreb* 
be opcrare con le g «,^ /. 'Ma,qu5lit>,chfi ifi?pofte plt^e aloK, 
nodopofattalapreuiacfperienza ^vofeireche quellamcdeii- 
ma artisliefil;^ facf fle vn liro propo&3^i, loago,'per ef^mpitl 
a 2 6 . pain . $i.cercaquanta eleu^otie. (k^iAdaf fi aJ pezao . 
OpCri&cosi. Seiipaffi iijo.j^preiiiaefpcirfeaka midan- 
no itf numerata'.lipafli3aou.chemidarannOi e troucrai va 
numcro il,Qua.!e.Oap^refaEpioJicIla fquadra afcrittoaflali- 
BfSi *■'•' ^idpiin-idiinffiieilzd^.I^a^iiglieiia.taiita cbciliila 
p^ per.]l)>iuuouw> & alIoi%il tiroFiufcira di paflr aaoo. 

It?rofi.iiucto.d*aziOpideitirifidannodafcIirtee'*»;*i*.i 
cperatierijiq|iantitddiqnc|lefipudfariadiie.nMi> ftism\ 
pervia(!Uafc]i)Io,'pcrche il quadrato "' 

drft£p?r4i<hcltapairatafigura)Sfempre ^ 
c^akaUiduc;quadrati,dell'alt£Zza >( 
tii UaHaqtiarta jatte idella lunghczza 
iS {StCOieiio>CQOiarcatuitelediuiiIc>^- 
ni°di^««p«lHferh.J,f ,v/,rnella Jstcfiii;. 
tc%ii(a}<laicwV9<<.legaklecirco- .,.:.>:. 

Cg j lari 



lari ie^c/,Ji^^chccmld[endoh'd$ dkiifagiiiii partf ai^ 
nutiffimeeguali»e(racHnifiveratu^ che 

loDoiteinpide^dri* ;' • 

CiB/effumo ftrQcheq$um9 diT^vfi militdre y/rimeMe tdtm 

flitudini , o lunghex3^ dei tiri^dre ^e imfottine \ e fidnd eli 

tnoUemmento .Vdtre^notutte eunofitk dccejforie ^ le qoutM 

fttuono molto piuforgufiodi Geometriaycbefer vtile digmerrA . 

Btro chi voUjff Ufquadrd/oDofer queBo ri/fetto delle SfnghcT 

ze i io ffenderei st/emicircolo z b c diotto^e ( come neUd fre^ 

fentefigttrd) ilqude hdueffk Idgdmbd ad, 

ccolfemididmetro e b diuifo in fdrti tni^ 

ottitiffuneti^egUMlifitcendo ilfrincifiodel 

Unumerd^joneddlfunto c.Hifiuddrei 

ututfii-funtidioUdfeftferU f» g, h, i , 

UUrp^ide ^h^ fi. fdrdUUMUzc^e 

€0s$ sl$dii€rehhero/ofrd Uch . diuifie , e 

nmnurdtetutte Ufti , g b » UqusU/eruono 

fvtU dtnfbtudini » o UnghO^e de tiri. 

rM§ULMfi/4ifmJlr4 f»4MigtMf/ i minutijtt fiHi3r49U 

. ndinm4ant€»gdudfsh€dumftint»d€Udnoftrd^i^4irMt 
-'. fh4 hi tfMntidifngndi ,f$n4 4C4ms9: 



I 




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K Per efeinpioiicer 
ca doueeafchila di« 
uifiQnt.del jfettiino ^ 
pumQOoftrbdifugua f 
le « Guardo la prefen ^4 
teTauola» dirimpec* M 
to al nuniero VIL e 
trbnochecaica£opi» f ^|**iy? 
ilgradotfi*e47.iia M\— 
luiu 4el 4|UaidiaQtie^: \\wi 
«idinaiio^ ^ 



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H9 

MA Gikihfi^ iimMUt9kJl4if4teUm$t§^t^p€i0 d/ 
fr$ia*i s Mem fi jmh ijf^JW' CiffCMfidm di fi^fingnt 
qmaUht cifd ciitdldKi^UtJt diUefit^i^ lera^JiHidiieeefifrd ie 
jfaftTficiertfifieMi^deen mdggierey erdcen tmmfdngeto itimm 
^UndXMne # . llGdlike ceniemfldtin^eie d^^freieaiin egni 
fMiue AclUiekmp^dlmlM^ e'hm/mra/eldmeifie'^lg4Me^in/lf 
mwdefifiiB y ciein/feii§Afnelfidne , JnsniferfemUcei^enit 
€gli percneitffe . . ^ ' 

Hotfufpmemde cbe ^tiimfeip memre 'drrind d fercmteri^ 
qmMntodfefidfemfretifie£eLyU cenfiderereme ^ e mifnrereme 
qmdnteeglifidr^fiiie dlfidne refifiemey n^idte ^ /Udmenie 
dalld dinerfitd m^^thgdi ddtimidenzd^fttn i BUdfdiere^tdn 
/# inefferies ilqudlk nen/dffid^ chctefdledeitdrii^eridifmem 
tre fercuotem in vn mtnte > hdttttefemfremittere, etmnmfert^i 
(ddid egn'dtrdfdritJt}fndnietdngele^deltincidettJodfdrifiit 
tfihdcute . Si cherfequelcdnmnecdnfe/fdmdUUredi ferrei 
e qudrdntd difoluere > mnfele sfittdd mddncerdfcottfUdfikcen 
Utir^ferfendkeUre7ktd§eriind\t^fendfoi U offtnderd^hen^ 
cbe dthid UmedefindidricdieUtttddefimd diftdnx^\ctatfr$^ 
ie^ene del tire > ch^effi. chidtndm >di flrifcia« llPfehimd^ 
ferfUdni^iefdffid^eintMie « Perifefifmdarrd ^ndkke^cefk ) 
memfufftfiente^ enonptardgeometrhd^eficemf^tiftdfin che *^ 
a^trdttktmj^.UtkitriiUye fi rifinti djfdtie ^che foeo im^ : 
fertd^ 

'-'Stmfefixiem'9 ; 
• fParlqeiio folainente peif i mi d^lParifglierie : t>er6 
Suppobia|no;e{ie ouella por^ionedellaliAea» chef^ lapafla ^ 
pocoprimaefocodopb alicolpire; fia come linearetta.Sd ' 
chefi traita diliiieaverainenceearu;f'> maauiefkto qutifta (fe ' 
faSt interzj la iua lunghezaatli (H|iditre mila paffi geometri^ ' 
ci > fi potra bene confiderame vn braccio > ouerbVApalmo fo- 3 

-8»:.StoppDniamo£KOipdat]am^ <^lc6Mf^^u€ttp im»> 
faidtfumati£U»ci3^^ i 

•w, i pofi 



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f 

T i 



i ^. Mfafe UoMidcfiiiia/paKkstN^^ . 

^)(^ebikio.dtuttfitca^ ' 

4iiLeiAp6n!aMttft&al^^ 't:t 

GsL P^^^^ ^^ * tempi • cioi Sc il medenmo (pxdoif$v . ' n \ 
4k m9tu i%ntkxi^^^matv^\tscn^ 
^quabiu tempo/. kt &)r2a deila prkaa £kci f;oiijQ*ikyS c d^la 

iecoixlacoinc #•. / . • \ 

>. 4» SuppAAiamo poi cbe tiiidiliri^ai^faidf^*^^^ 
10 j^fe ftc(I>>. femprc iIinedefi(nQ«|ap^«. iicli^&«^ 

guifcbbe^^i^andioilidoferma ITaiddgftcna^ipa^we 

liiel medcfiffiO luogo > cdn la tdedete^caika^m^ 

4c£noa elckjazioee» ediftaiiaa&c«i!l(il^^ 

Pobliquicadel moro «. ,-,».. 

. Supjpoftocwefto: inenarin^mp^ 

a^mwo i^^ftot ia libea, e diritiufadelttrpV^^ Mipcbdic(H 
ll^e^^l nmro » o nd • Se ^pctpcDdicoIdte» J» pOKol£i t>pem 
cdnviia tal foaa. (^ p^oocremo-eiilb larmafflmchepol^ 
auec quel tiro .} Se iara ad an^oli obliqui » come la linea sh « 

aUapaKCtciitrt Jonotocheritpccteattapaiidfe ir »foiio od- 
lalinea di delproietto due motiiniie 
me compofti: vno cio^» di auuicinamen 
to perpendicolare alla paroc » ^alfrodi 
p^agg|olat€»alejOparalkk> allaiik^ / 
ia4Ip^i^pcndtcolareciviQu$i€,nKrffar^ . 
to » e milurato daUa linea* 4 r ^ il par^dlel A 
lodallalioea r^;poichene^e(ij^deiinMr 
tempovengonopafladdallapaliaambi -. .: ^ ^ ' 

Hora otferwamo che di queileike foatdAnpisto yvnatfobi 
^acpropoiitOy^^ccrelberieibtaicni^ettioaixx^^ IFmft. 
iOt& inteinar kpal£i ioeiso» cio^iMJ^^ 

pcn* 




«41 

penMcoUtedc. Vzkro^Mc^utdxcfijffcixAmto^ mn zccrc^ 
fccra tnai la forza del proieno cpmro aila refifteoza dd mueo^ 
§c per6 n on gli accellerafle anco la lazione perpedicolarc. Aik 
21 le fiifse Porizontale ren)plice»efok>> ienzamifiura alcuna 
delperpendicolare» chealtro£uDet>belapalla> fenoncorrere 
cquidi ftante dal mnro > fenza nuu toccario>.non cbe romperlcb 
fe bene fliise vn fotttlis&no vctrof Quando dunqqe.data la di% 
rittura di qualiiuoglia proiezione> noi iaprem(>quanto di que«r 
iloimpeto peipendicolareentra nellacompofizione del mo>» 
tOtfapremoancoPatciuiti^omomentodelproietto verfpia^ 
refift enza della muraglia comrapofta f 

SialalineadjqiiaUiuoglia in* 
ci^nza » ^^ r fopra il piano ^/^ 
ftcSa cott qualimaue indinaiusio 
ne, ma pcrd fia la porzione dt 
tanto piccolache pofsa coniide«> 
rarfi per mta* Tirifi perpendico* 
larealpianoln^^o eficongiun 
ga c6. Tanto dunquedimoto 
parallelofaranellalines «^•n£> 
pett6aliaparetei[/l>qu8ntaelalineair^« MadiqttcftonMi 
6cciamo ftima ; perche moltiplicato noo aittta» e d^oiiito oQ 
debilita il momento» mentre l^ioro in^to npnaltentto refU ii . 
medefimp. Dijpeipendkolarc poinelIafie£«t.f8r^ffl^ . 1 
la linca 4C . e IdiOTzadeicolpoi^aciagioiACiQi^ 
do cbe nello ftefso fcitapo iarafcoc£ila jtf^ nuggione O pu« 

Suppom^amo hora che U&nBadeiifiddeju 
Perfaper lalbrzadiqiiahtnq;altraii^ci<}enzai/^iprei|da0 d 
l.egualea ^^,etirataia i^#;perp6adicolareai^ano«farii$ 
forx»adiqueibinddenzacdaie:l?&ftdkar^^l^ Pokhe.^ . 

4^«i/^.fbnoegaaUfCibiioitiixddKfli^^ F^fj^j^; 

nofcorfenelmraefimDtettipow Adunqiieanicolef <> 4i<«f0.7 J^tHh 
no fatte iicU*iftefio teqopo; per6 gUiiDpeti ri^^ al nnroibfr 
oocotaeMf.de. 




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f*mtMe ik$Uiii*ti fm»*mt ifetfhutidefj&img^Uj^.fUiime 
4enM, ■■ 



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\ caut cb qui per CoR^Iariache laihcidea la pffpendico- 
kre ^b . hdinaggior fbrzdtitrt&terlraiiEte »' elae odo la foiza di 
eCsaconieillfehototale«..E(sprMe£kyae j^alleia .•nonneliil 
f^eMefe6efi4olafor2a6ia;:coafeienaitui9ar,|;ti^ db^ 

i^ aimofo di 31 o. gradi ha la meta.della£bREa.ro{ale^fseado il 
lenarao bntetsldel lemidiametro'. Le altrep(»> cooformcaiie 
rannonu^giore ominor fenprettov aueranno' .ntjggiore , a 
mihorfbrza* 

Le foize delie prokzKicmi hahno re^ipcQcanicniiP ^ txiedcr 
(ima propor^ione , che harnioii faatd«bitt«igolQ«' c W ^ul pi^ 

fiovienibnnatodallelineedieil^idcoKCwl 

Siaynaproiezionefattaparralinea . 
irr l'altraiH?rfah'nea<r^. Efiailpiaiio . . KX:. 
del tria&gclo dhi» peipendicdboo kl ', . -> a 



muro i' 



5 .- 



•<^mpd dT§ e k» fpauiy 




^^j^ .dk cioi Cfeuaio il moto parallcloj lo 
/'i9^/~'fteiso^finzio-X'r(i;cdrreiieiteafpo4^. - , 
i^* &rafiiio leJEbfferedprodie deitempiw. . ; i. •. 

CHdfclaibrjBaper^if .iari'caikiC(ii'-bpcr:<^ fard^ome df. 

•' MorMproiettr>iwct{umavIa.itti^fii?«i.xnelp«rcuotcr^ 

quando gl^petifaranno comc lc fecanti dc gli angon del 
cbfnplementodefleifidabae« . ? 

^ Sul^iti^ietoperlaiperpeiidlcolare^A 
<!;ome dS < & abbia' vnam fi>«Ea ^cdo 
l^mpe(operi'kcIinab.4<'*4bbbi<kfne« 
..•.. . demaa£3rzd>dk0cttbnn)petapec«r, 
alPimpeco^r^j^deoee^^ecome 4r 
alla 4^ ) laquale d^ ^«ibcaiucd^^angola id^ conpleme 
to deU^iadiaazione* 

•'N^ Poidic 




fcfacannogPimpetipcr 4^, & ^^* comc fono gli 
ipazii di, & sc . imobili fcorrctannoucimedcfimo tcmpolc 
duclmcc 4*.4^.cio^loilcfsoAUuicinamcntopcrpcndicola- firisi 
rc rf^.Dunquc.aucrannolamcdcfima fi>i2a controalmurot /w««^ 

Dipiii fcxol talcxannonc» c pcr la li 
nca c /ja pallas^intcmafsc tutta pcr Pap 
punto ncl muro ; jidunquc pcr tuttc Ic 
lincc clcuate Jion folo s'immcrgcra tut- 
taneliafolidita» mafiira femprc mag- 
gior pafsata»perche ha maggior forza . 
Ma dcllc meno elcuatc > pcr chc cifcuna 
auera.minor forza » niuna cnorera total« 
mentc.ncllaparctc» maalcunc ancori* 
lalteranno» c sfuggiraono<dairiiraapartc* 

Sidftrbd€tt0tMtt$qM€jl$.dBT4^nd9 ddvm Cirt$ effctt^ tU 
fi€gdm€nto > o rtfrdxjMnc chcfdnnoiwoictti nclfdffkr con in^ 
clind3(jon€ ddmcKXMratodlmczxofiudoilfoJncitrtumdofildii 
ncd dconndfiodclUrcfrdS^oiudcUdlnfCyCffc^c vifibUi,. 

rJNE 2)f* LIBHI D£L MOTfi. 




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BE DIMipNSIONCt 

P AR ABOLiE. 



Solidique Hj^erbolici 



TKOBLEMATA DVO: 

A N T I Q^V V yit^fif%, T B R V M 

In quo quadiaturaparajbolae XX. modisabsofaiinirj 
parrim Geometricis^ Mecaniciiqae i paidm 6X 
iodiuifibilium-Geomeata dedudis C 

radoiiibus .* 

fj y y M AIT B RVMf 

fy qu9 mirMiscuhfddmfiUdi di Hy^boUgtd^^ 
dceidcttiid mmtulU dtmmprAntw • 

CVM APPENDICE 
De .DiineQfiooe fpfttij Cydoidatis,&Coch!c«« 



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LEOPOLDVM 

AB ETRVRIA 



a: 




IFFICILE reor|5circni& 
fime Prioceps LeQpolde^fer 
rea hac aeute libros confcri^ 
bere y difficiUus dedicare : 
quandoqttidem booarum Artium ftu» 
dia vbique ia belU degeuerant > & Re» 
gnances viri oon exigunt ingeniorum 
vires > fed corporum • £truu:a tamea 
Begia^non minus fbecunda virtutum» 
quiam Principum^undum cdocet^eaa 
dem efie Mineruam & BeUona> vnunw 
aue ApoIIUie > qui arcum fimul amat ji 
& citharani • •SerenifHma CDim Celfitu- 
do Tua ( vt rtf liquos oniittam ) littera- 
rum)& fcientiarum omne genus perin* 
de fouet) colitq: ^ ac fi mundus alta pa- 
ce &ueretur^ pulfifq^JFttri;s fblas Mufas 

A 2» domi- 






:4 

dominarentur. Verum aliamemator 

difficultas tcrret ) dum ego tenuitatis 
meas confcius mecum ipfe cogito^ libel 
lum hunc ad eum Principem ire^ qui il- 
lum non iblum protegere poteft > (ed, 
etiam iudicare • Q^icquid eft, non acre 
iudicium Sferenifs» Celfitudinis Tuas > 
ied incomparabilem- humanitatem in- 
uoco^illam inquam humanitatem^quiae 
nuper amplidima in me beneficia con- 
tulity & humiiacentemerexitfortuna 
meam • Audiat preces meas Dominus 
I^R^nantium^talemq; Principem diu 
* cuffodiat : fiquidem diuinitatis interest 
huiufinodi viros proiperari ^ vt aeterna 
Prouidentiamagis elucefcat9& coniun 
€tsLm aliquando cum poteflate iapien- 
tiam in terris demonftrare valeamus . 
Serenifs* Celfitud. Tug 



HumiUIinus» &obrequmtif& feruus 

BudHgeUfiA TtmkelUits , 

' AdLc- 



AD LECTOREM 

^ Procmium. 

» 

* 

NVllus ih tmiutrfo Mdthematiurum difiiplsnarum 
Thedtrofirtdffe tritior puiuis reperiiuTy ^u^mfo* 
raholie qutndraturn . ^uare er^ (intptis amice Ltm 
Hor) drU tritum artumentum t^m diu defudnfii f libenter 
e^uidtm excipio oileihones tuas ifed^vtimm yitimus defi* 
iiauerim • ^am tamen <veniam mihi negas ^fcias eandem 
piurimis j (g/ eff-egie iaudatis ScriptoriSus te denegare • Olf^ 
ic^um enim de parahol^t quadratura , tptod noSira hac ^eM-' 
t€ con/Steor mihi nimis iam inueteraffe ^ crediderim neq^ nom 
uumfuiffi Cauaierio ^ Gaiiieo , Luca Vaierio , (^ alifs,J^iu 
immo ipfum Archimedem aecujat y quicumque improbat ht^ 
tui/rationes circa fitineShm yetus infiitutas . jiudiamus 
ipfitmin Proimto^nadratura paralfoiit/vhifcrikeHs Dofim 
theo inquitn Eoram enim> qui aatehac Geometita^ 
operamdedecuDt,DonnuIliia iDuefiigare» &meaio» 
rise mandare ftuduerunt , circulo dato , vel circuti por* 
tione quacunque , fpatium redilineum xqaaleilii po£> 
fe inaeniri. Itcm fpatium i coni totius re Aanguli fedio» 
necomprxbenfum & linei re^i , z4 quadraci fcrmam 
& menfuram redocere conati funtiiiimentes nen faci* 
\i. conceflibilia fundameBta . J^uihus fverhis diftrtijf^ 
mefdtetur Geometraru Princepti argUmentum iibrorum Dt 
dimen^neeircuii , O^dtfuadratura parahoi^ , nequefuum 
fuifefVeque muum* Sedfi quis attentc confideret Proe-- 
niiaUm cfifioUm > libro de Uneisfpiraitbusprafixam ^ intei- 

iim 



4 

ft >i^ maghiextArte Comms-J^^liUximJt emm Trope^Om 
msUbrorum De SpherA5ftCjrbft^l<H^ De conoidibus, 
& fpbieroidibus j & Delineis rpiraltbus(^£«i lihri imer 
efer^jMcJiiifkdtt PrJficipam. IbcHm tmtf0)iCmu^ fiim: 
Qui (*vt in(jm MSkoty no« (adatQQifOf i»ai&luec eaco» 
gionda rortitus, vit«D perinutautCk & ifl^rcli^it inez- 
plkata^cumiUa ioueniflet, & alia quanipiiiriaift ^fs^ 
qujfidet» afi mnlttUD ade^ Geometricas ucultarcft ao^ 

pliaflei . S^i er^UcuitadmirahiUi dcfnfeditiiva ^ M f t m i » 
circ4 atiorum imentd Uhoraniquisnegdlfitiffiofiendmm itu^ 
g$miit9 meo. mtttuatd thearematd. ctntempUnti i Sed ^ 
4juod comiufio dnti^ajit 5 drgumeatd certiifuii>us HU ceum 
prmd^itttrtytpkrimumuouderunt, ^ induditdi Imm» 
cttmaddkerdmpdrtemliheUi dccedemusy m qui defidida- 
da^ohypcrbolieodicendumefi t mmfoUm,iffum Theerema. 
inexcoffjtdtumt0*fvtitddicdmpdrddoiticumeritifedetidm 
demon£handirdtio inifitdtd,^ penitumoua * Verum f i«w 
^is) reUqui Scriptores, qui huiufinodi quddtatufdmaggref 
fifimti yelpnguUs > ntel ddfummnm hinds prodideruntsue* 
que tdmen mediocrem Idudem confe^tifimt . Fdteots fid 
nee ego UbeUumhunc ex prxtfeffo inftitui j compofuiques. imm^ 
quodt^ dlifs, mihiquoqtdcdditilinguUs hafie quddrdtUm 
ras diuerfis temporibus inueui , quds in. ymtm odkBds wtne 

demumyolentiiusfimuiexhiiiep.Tutdimm excidmdss Ueit 
nimis efi : quotus enimquififi nperietuniamfiimeUeus Geom 
metrd,qui legdtpene yicies repetttdm propofitienemi cum nu^ 
nuro lemmatumfere dupio ? Huicfine oine&ioni Ithet copm 
tradicere . Cum emmUMlusiuFt^faiotfes^ytplMmmtm 

non 



i 

i^mmtimtimfimi^iffit i^ fitamjiiummm Mmink 
«•««•< fibicaitft pNcetar 
ikker jt ; dkuAi A fic brette fiet opttSv» 
• ^i^ftfimmsfnltittteo»fti»mttmmqmy»*m,iittd* 
ttrJimt4Mmm^tif4t»f4mtiiierei fmsffthihtt f f^ itt 
bt€ iemnftttsTnmtmySMaitef4mifitame»hocq»op aimit 
ezdJmitmr , mHtm,'. i^tiiittttm txi^s ^ cettceio s &* i» htne 
partem liheUitm excu/are non attfim . attamen non ieeritfot 
ti^^t&jmt^ftmitiuwttttikmnon e^hflititet , cum Get^ 
mittriau^ft, .SalatmmC«mePtiaimtr4iheraiesilifiifli'>t 
muammertteiumt i »g txiilim > iimmim^ rtiiit aicimtom 
ttt wormmiMt mftte^ im^efltiefmieniaft tittMs enim 
pm^iku, imgmiitk^^i twmiptt»0 i$t n» GtmHrita pait» 
Mnt^ fteiiinktJfi»iiim»ft!^*xtiriitrohiirhaherefiiet:prjs<^ 
fftthmfifimftr :>^Qr*. mtitteikt , titcafltulut jtr^iHtt&iiT^ » 
rti iftmutynaiuitttpfiHttiimiscirci Jtritmttieam^ artem^ 
tmfttmm • *mttt ttt/ut dmitKtmmtificmm iepeHiet , rentiirm^ 
mtF'» ^^ttetiam arci mmfittia fimi trmn > Q^ aipUtunii 
flaj^ ttt u inm » V9itM»kifimiiffiafereifit$iai^/me iamna % 
fiuil^^tifoMSfn^imft^ timtitmeimeiieOafiurit hmtfm 
mmhrt nm tt at w u*. Sti^^iimuitit ptnitnshiiteatith 
li^tHmsjfiiufiiia KcfMii6. mhii imtn^ M^*'^ ^jmiirat»*' 
ra^fiiu,fniamtitiMiShimfiuniitt^et^taMfli&»t, 0*iim . 
ftitmfiPtia * Hm, ttet^tA iviStm mffimitii^ Ktnerenii/s, , 
l^ Sim ii 0$ ii C/^k&ttt iiit^iitrtneus •• ■ipfienim iicet , . 
efiiifi^riniipts terra yfiltm iihtt^ iNiuteWi(l£^ apparen* 
t^iMimm inAtmhimmiifftifiimfti^mgi^^iiam 

iiti^ 



«»^ :i<« ' 



M4mt4tilnM Pi^ores ; Mitjtci , CiiUrAii^ Pdets» Mt^ U 
imifiH •. Coiftfarv^A dif49Ji4!ffittt,atfi ofilms yjoffic^fqi 
omnibus demerendipidoreSiftortun ^&tatimUU aliapar 
efii cattpones ifittms^ ^ fticttnqttcammcotunt^-vit^ebg. 
mmtmJitmmopeneraitiUm,. ^Hiaetiamfi^vtdttaisfila ^/- 
tend^tttr^Ja^naiidttseritidni-vftu t t^deteHoMU udtm- 
ra^pittearitm . AfinfMmmofratiohabendaafttaitmiusr-' 
$ilitiieest2m'faci^ efi nmerm» qttam digieilefit ijs mttm. 
<dijrefe-. 

ytcftj^qtteeafesfifiha^dt, itewamits ad ohieEhoatt 
fy^circ^itrtisfitndamttfta^verfintttr;, Uidignor tfnidm 
l¥camValerium:,'veren^ifatttUArchiimedem> fumop. 
timamcaitfamfaffepiptjfejfimd definfibne riffitmfit^, 

^itnt.Aemdit(s<Mlpam^trat<mmGe9i^cartimdi^ 
fi^h^ Mtsa^ttisfitvdamtmf imttxte Habiiiimttir/am 
^am d^htM»*^ fuppsn^nt : .akentm,» qubd fuperfi. 



qubdfilaqtMemignitudiDes adii* 
l^ain&i^eAdum«(|iudiftaiftta%poauiiciir, cumca- 
niep iq cQmro terras cQOcurrcj-e debeant / ^^'civi^ /« 

^«fumfentemJM^ytlmtlUmtxbisfiippofiticnib. ^falfam» 
1t^lveltqit.^onini.aprincifia GtomttritefalfiitxiBtrt tndtm 
tnodo. Fa{/^nim tSi y ftty eiri^ks bakat centrttm, 
/pJf^</^r^m,cems.fiiiditattm, Laqitordefytrisai. 
firA^ft^l^Qfiimttrittcjntfidtrarefoletsmnaitttmdtfi.. 
ficts ,^coif0^tif ., Ntetffeigititrtritfitteriquhdcirciilicem 
^^^f^erficftsffhkrM^fiUditMtmy t&f rtiiqma hmiif, 

nmSUmfditimbithkaitteiciBtmtidmt 




&»m m Bodem pmfis mdegrMtfat tf infgHris Giomt- 
tricis^ ^omodo in ijfdem eft centrumj perimfter,/uperfrciest 
JoliditAs ^c, Lnuddrem i^itur in Mecdnicis. contemplationi^ 

bu s noua defmtione fimras generare shoc, aut alio non. al^Ji 

mili modo . 

Quadraium eft quadrilateium ^ quod, cum «qujla- 
teruixi > & «quiaogulum (ic^ fingula ipfius punda roo- 
mentum habem pcocedendi verfus aliquam mundi pla- 
gam per lineas incer fe parallelas • 

Huiupnodi enim definitio omnem demeret occafinem dum 

hitandij illis , qui Mecanica Arcbimedis opera)fecmdim ip« 

fms mentem non acciviunt^ Sed hucufyue di^umjit pro o^. 

literandafrimafiijttatispota, quodfyir^t Geometric£ ffA» 

eees/int, : 

VetdonuncadficHndum(ytaBqM exipimant)f^fim • 

• "Principio > wtUgatiJlima efl etiam apudffrauij^mosviros oiu 

ieiiio illaj videlicet . Archiinedeai luppofuifle aliquod 

falfum , dum fila magnitudinum ez Ubri piBndeatittm 
confiderauic tanquam inter (e parallela j cum tamen re 
vera in ipfo tetrx centro concurcerie debeant . Ego yim. 
ro (Modpace clar^Jmoirum ifirorum di^umfi) crediderim 
fi$nddmentimMecanicumloneji aiia ratione^confideraiu, 
dum, ConcedoJtFifia m^^Mtudints adUbram Itberefifi 
fenddtttur i quodJUa nuLteridiaJitfpenlionifm amuereentis 
trunti quaniUquiden$fi^ula.ad centrum temt rejpiciunt* 
Perumtamenfieadem Isbra , licet corpcrea , eonfidmti^nm 
ittfuperfidetemtyfidinaltijSmU regioimbut.afltrd erbem 
SidisjtimfiiaCdummodiaabuc ad terrfeetorufff refiici». 



15 : .. . . 

diSfdf/tik. Coneipidmus iam ipfim tihdm Mecamcdm n;L 
tf2BeIlatdmliirdmfimamenti$f$ihJlhitam diftantidm effe 
froue^am j quis Hon intelligitfitajufpenjidnum iam non am^ 
fliiii conuergentia yfid exaBe parallela fire i ^ando eg& 
conjidero hbram ^figuras Geometricaspondirantem ^ non ^u- 
cipio- illam ejp inter cartas iiirorum in qmbus depi^a con^ 
Jpiciturinequefipponopuni^um^adquodmagnitudines ip* 
fius tendunt , ejfe ccAtrum terrd, sfidtthamfingo ih infinitu 
remotam efje ab eopunHo ^ ad quod ipjius grauia contendunt . 
Sipojlek ibi cohchtfiro triangutum atiquodtriptum ejfi cuiufi 
damjpatiji retrahatur iaia^indtidrieipfi tibra ddnoftrdsre* 
ffones s concedo quod retraSd tibrd defiruetur dquidtfldntid 
fitorumfiJpenfiomJ:,fednon ideo dejtriieturproportk iam ie^ 
monfhdtdfigurarum* Pecutiare quoddam beneficiudf bdbet 
CeometraiCum ipfi abfiraSJtoiifS ope^omnes operationesfiias 
mediante intettefiu exequatur. ^isigitur m^i hoc nega^ 
uerit^fitibeatconfiderarefiiuras appenfis ad Ubrdms, ifU£ 

^/ ^.^. onpntumin infinttam dliiamia 

sfais^^f^dlbebli confiderare tiiri 
mfi^erfidie terrk connhtiid^m^cmusumh AbtirASld moffd 
ikdhies tetidiLnt^ non aMe^mm tirra piii£itHm,'Jeiidakn^ 
fr)iif^:edni^itijeypeWM}o^^^ tridJiiutdWpiratfoh» 
ifjitheoeiidfnjfhj^aj hhli^^qiptomeirtH, cUmnimii^fefp 
bWek)a%dtUs ^^eia^iiiin^ mh)ff/<ms'aaifftits^^i^^^ - 
aaSaiii^ifii^ir^utncMil^ijaiin^ etM "feHepc^mi 








» * 



fitis,qu^ verd/mtti quemtdmodumfant yer^ft0o^ffigit^ 

rdrum,qu4eindefinitiomB»s4dhihenturi yerdfti^m erunt 

qiMcunqiThecfemiUd per AUcdnic^srati^s 4^jj^jfdj^Ifr4* 

bewtibus faerint conjiaeratd^ ^qup fer 0fdf fpj^fnes icr* 

monftriAuntur , Tuncitdquefdlfumdtci^teritf^n^diiffu* 

tum MfiCdmcum, jfempefiU Ubrd^dUeU c^ , f^d^do Vfd- 

mitudines ddlihrdm dp^nfdfifadfinti redkfque,0' dd ter* 

ne ctntrum conjfirdntes . Non dutemfdifam erit^ (jifdnd^ 

tudfftitudines (jiue dbfirdBd ^fiue concretjcfint) ngn dd ce/fm 

trum terrd ynequedd ^$udpun^um frppinqu$m f^fra. refi 

picidntifeddddliquodpun^umin^nite dfifans conjtitjm* 

tur* . 

Cdter^m tireuiidtis), fffdciUtdtis trdtii i <vp€dh^^ 
cofffuetis non diJcedei^usipUn^umauemudMiqmdmdj^ 
tudines librd contenderefufpomihtur, Centrum^terrdmmi^ 
nahimusiP.ldnum yera tUui, qmdereBumeft ddjinedm com 
neSentem^ddiciHmpufiSilftm cum centr^f uhrd >. Horifmm 
tem de moreji^peiUhimus . 

i. 

gnitiKio Uberriofpcod ex ^j^ualibec iui.puodlo 

Qunqjiaipquierc^CAili^JMic^i^lcnvi^A^ ad lofii- 
mum fuae fpbSMp^Q.duQ^aa^enenc . 

CmipUmusfigurdmAoC^fuJpenfdm exfai pun^o D^ 

B .2 medidU' 




tkeMmepdEbslMhUcili, ' 
ita n/t in quamcum^ue partem 
comerti poj^t . Slt centrum mra 
uitdtis"F. 'fonamufqUe reUam 
EbG ,per^Hdiculartm ejjt^ad ' 
hdrixdntfm* 

Certum i(l ^, donec centmm 
V^fuerit extr^ perpendiculum 
iSGyfi^u^anflp/amhum^Uam 
f^dii/uiram ej/e . ^ando^ero 

puh^um V.fuerh inperpendiculo/ujpen/tonis EG , tunc fU 
guriomnino quiefcet : Centrum enim grauitatis ipfius nuf^ 
quam poteritamplius inferiiis dejcendere : ^tn immofifi^ 
^ra moueretur, ^entrurn ipfi^hi afcerideretyqi^odejfinon pom 
te/l.Si quistnim centroE^mteruallp EDF.tamquamnmi 
reda lihea^Jpb^am concipiaiejp dejcriptam s ipf^ eritjpha^ 
iray in cuiusJuperfkiefereturpunSium FyquandoEDF.exten 
/a/uerity &*adre^itudinem redaGfa . certumque e/l infimu 
pun^um huiu/modifphior^ ejfe inperpendiculo EG. 

^c|Uipohdera«cfibi ipfi (igiira dicetur , qux ab ali- 
quo fui pundo liber^ fiiipefiia maneat, & ad nuflam fui 
partem inclinetur . 

IIJ. 

; ' iCqUijJotiderat nbi ipG figura ^ quando (^cam liber^ 

Ibfpenlafit) iq ipfoiurpenfionis perpendiculo cefitrum 

grauitatis reperitu^ . fi enim adhuc mouereturiCentruni' 

grauiutisafcead^ret. Q^eftimpoffibile. 



Cen« 



IV. 



IJ 



Ceotniin graaitatis tnnc reperinir in ipCo f ufpenfio. 

nis perpenclicuIo,quando figura lib^r^ furpenfa fibi ipfi 

aeqttipooderat. AliasenimAguraquiefcerec, &cen- 

t nimgrauitatis ipfius pofiet adhuc inferi<i$ defcendere* 

Qpodcftabforduoi. . 

V. 

.Ceotralit^^adillttd libras pundum appendi figora 
dicetur^iaquodcaditperpeodiculuai^ excentro grai- 
uitatis figura? produdum . 

J^o emm Ubra AB, cmus/UU 
crumjh Cy^ dd ip/am Mppeiu 
fijftfigura CEB , itd i/t totum 
iatus CB cobereatf ^(ifvelu^ 
ti adipfkm Ubram conglutina- 
tum^ Efiocentrum prauitatis 

figtrapun^um Dj^ ex D agdtur perpendiculum DF adbo» 
ti^ntem eredum , . 

' lamfigura CE£ dicetur , €^ confiderabitur ) tamquam 
appenfi centralitir adpun&um F . • ConSiat enim ex pradL 

^tsy^uodfifigura iatus CBfi/uatur yt/di^s i brachioUbra ^ 
filumque remaaeatfilum counexionis 2>F, nihi/o tameit mi^ 
nus ^^a adbuc maaebit ^ptius manebat ) eandemqi/er^ 
mai^ werfitsldnram pofitionem y quam aniea babebat • Fidlf 
jif€h*^rop.6»De^uadratttraparaboia» 

r.r • • • VL 

. . . jEflttialia giauia ex zqoalibus diftaoti js a?quiponde- 
lant > uoe Jibra ad horizoatem paiallela fuerit) aoe io« 
4lioata« . . 




• ^ 



Etgra- 



£t grauia eaodem rec^roci racionein habentia» 
quamanQaiitfc,a:quipoiideraoC|fiiie liknfi jSc.ad <bori- 
lontem ^randa» fioe ioclitiatt f 

dem in ticGktM aqmfoudft^uuimiMmm^itiiiifitffdmUtrJi' 
h^dhorrl^^nti xqmdiBans : AttMtm fm* fSleiidi fefiint» 
non ommittendam cenfio detimi^atiomm ifr^firtim am 

mJm^lU ex&hrd mmerudi mdefidfrientd^ ^ff'iw>iniji^!9pf' 
nmt^^inifaelle^mittdm^firiiif^ 

Efto inclinatA lihrA 
AC fiifienfi ex fun^o 
BndfilumBD, Sintq; 




magniitidines SF£ j &* 
G. centratith aofenfid 
ex pun^isEj t^AEt 
ponatur effisVt magni- 
tadoBFC^adm^gmtU' 
dinemG, ita reciwoci 

^fiamiaABndBB, Gin^hum M.»Jfiimmdsim/ind>. 
Um , magmtudineffte^,iffifbmUnin^ymt^^ 
re ^O» ^equifmderare * 

^roducanturtmmpei^eudiadaGJN yh&l^tentm 
ffomMiffigirarum G^ &*Z, trmfamtiuy-dmmm^i imu 
f^MiihraCHyfmtitemi^fiiJkmifiiiim.^^ 

niamigiturefiferfiiffcfitimem^-nm^tudomc, adm*. 
gdtudinem G» ita reciproce AxBudB Esfiue Od^ paralielasj 

HMadM U ^fidponderakmitm^egmudiMtJiBFCjttj^G , 
4dUramhoritffntdimHCtfpa^x. £i^»mmmime4ieamM 
C^^f^ffritQmainoittperpendicuiQDF* Proptemt 

^nitu» 




»5 

gnifudines tqui^nitriAum ittMtn iumsd Urm A Cfiifm 

fenduntKf : *Uh ^pmuertnm yCBmmum Mmrum gr4uim 

tatisipfirum, quod demdnflrdtum efi efi Juf€rf<ndiatk 

I>F ^pfienderet , ^uod efi impojH>ile , 
Hdc autem bremus 

eomiudenturbocmodo . 

ComieBdntur (in e«d$ 

fiff^r») tientrd gna^u-' 

tis du&d reBdCft* 
J^uonidm ma^tudo B 

F C dd ma^nitudinem^ 
Cefirz/tABddSlB, 

fiue ( ob fdralUlds ) yt • > 
GN ddN I) erit N centrum amnune ffrauitatis mamitudi- 
num appenfirum . Si ergo librd A C non quiefiereti centrum 
ffrauitatis N^ afienderet* Cum enimjit in perpendiculo DF^ 
moueri nonpotefi ^n afiendat . 

Non me lattt Au^rum coutrouerfiam , circd libram itu 
elinatam > an ftdeat « maneatue Jufpeuere otntra magnitit' 
dinum in ipfi libra eje coUocata . Nestamw , ^ia in hoc U 
beUoifimperconpderabimusmanttmiiaies irfia tpfim U» 
bram appcn/as , nmlkmiusrei moBrtd^reiiif^ > quam aliorum 
cotttrouerfi^t demoifirMifmemmMaii^aPei 

Cateriim p^nesparakUufifis in optkisproorelfifippo^ 
temus tamquam notas , y'Uitj^^poi/onij eruait} njelArcU 
medis , njelfidtem e:c ApoUonio ipfificiU negotio dedtKets^ 
tur, cmufinodijiint ba, qu* fepuuntur» 

Si Tarabola re^am Uneam tangentem habuerit,i qtabuf 
UbetMitempunffisipfiustangentisre^aUne£n/que adpa» 

rabo- 



riJnUm<lniutt4atiir4tqmMHmUs£iim$trt,trmt iemtl^ 
fiinter fe Imgintdmt tt fimt fortimes tMgntis potattii 
itUtrfi. DedudtHremmhocixio.frim. Comc. Nimrc, 
BtilUdemtJptftrtimilms diametri refimdents tt fatrtet 
iffitts tangentis » ordinatim afflicatis tt^ualetfitttt . 

Item ifiintri farabotam i funSits qttihushhet re^jtiL. 

biu ordinatim ditiia , qme ba(is faraioU dicitur, reiix Ustcat 

triganiitr diametrofaralleU . Erunt ereQtt iitterjeyt 

pott reSangulaftSa i fortimiiits hafisf iptit 

iiHffis ereSts atfiindmtur . Heceiam 

0- i Caualerid )&i iuhisinji~ 

tiadoliliroJemotutHeit- 

Jitur. 



m 



QyADRATVRA 

P AR ABOLAE. 

Pluribus modisper duplicem poiitio« 
nem>more antiquorum, abfbluta. 



LtmmdFrimitm; 

iX PARABOLA duas tangentes habuerit l 
^ altcram ex tcnnino b«fis,altcram vcro cx vcr- 
k tice;tangcns,qu{ad bafimeft, biTariam fc- 
k cabitiirabilla,quzpcrvcrticemdiK:itur. 
S tfttftrthiiU a b c , cniHs diameta b i , «M^ 

Mtimvmdfflicatd(fiiKbtfii) fitiHit/a. 
gtniextermimhitfisfitdifer 
verticimveritMgensbc. hi- 
C9 if/dm c d . bifdriim /ecnri i» 
fiiiiUoc. 

Ciimjt.l: ifit tilngiiu,i i didi 
metaterimtdqudleiiiiterfeih. ^ ,^ ., . . '^•_, 

bi. Ztiuin^iir^mmiiffli. ■ ^ ^i^ 




1 8 De Dimeniione Parabolae 

'»• rcM^m^ftrtmftiam'6.t,^C.,Sifi'detdtvJle»deii(UiUmH^, 

ZfmMM II, 
SlfaraboIaduaktangenceshAbuericexWfistcilninis; rcAa 
lincaquxaboccurfuduarQmtangentiQmdudtut diam^o pe- 
raUcla,propofit*parabolsed!ametcrerit. 

£^0 pMTMhU a b c , cuius ex 
fititl^is ax^ cidnte i^g^tfftjrsJ^nt -.•^ 

zd^iC^djroMft/rft/ttes iM 4\ Sm-' 
ftinSto MMtem d ) reSM dMCMttit^ 
AcdiMmetro^drMlleld. bieeif. 
fMm d tfrofofitMfMrakeU di^t 
pmmeffk\ 

Sit eMimt'Jtf<^ile i^, dtMme 
ttr f g . irKat erge ob tMagehtem S 

/cwwVo ^^Mqudlesittterfedimemfifr^ 
fum. tic»es{h,ho.lrerMmffinihgett 

tem c i , MquMles eruht i b , b g . Et ideo M^udes erunt imterfe if 
/m fb,b i : totMm,^fdrs . ^dftri notfpoteB. Noa efiergo dU 
dtMmeter frMter iffMmA^ , SiufderMt oHendendiim <^c. 

temmM III. 

Siparabolairestingefflfcshabnerit; duiiis 4d bafifli, ifcttt'. 
BftnpervAticem jcrittriangulumfufa tahgctitibttscoR^nehcii 
JtiAbA»phtriitriianguIi,quodonruresdU(%itiuarta! taogemis 
per vertjcem alterutrjE fcmiparabote . 

ffiofMTMioU abc , cuiirs bMfisXc^ diMJHetfr b d ; ^a* rJo^rt- 

tesMdbMfm^^ce:.TMngensf^*t/tftie^fttih%.^emn*- 

"^ tnr f Arf, corKi&futangentiHm a i^oi^reSiMi \ , diMmttrafHrdr- 

fttUm. leU:eritqMe iididmFterfarah^^xh. jDtrc^aydetti^xi^int 

f c g > y** tangentiiMs comfra&effii)» , o^ttfhiftr iffi fttirati^ 

Ifm, 




xjk 







Imngdfitr ab tdfis fdrdBo^ ' 
l f ^.ih .ertintq; pdraf/fild^zb 
ivci;& c$imfintdqsfai(tsi\y\z 
cb sangtntem a f> mt ^dtifU 
ipjsusii ; ideoqne^trUf^ulMm 

a f b quddrstflum tridngfil^ fi^^ 

hi JhniUs 1 f nv. £rgo, etiin^ 

f be qnadrttfUfm erit trittngn ^ 

li If m (y^/K enimfer Ltm.fri 

mnm dq^Ales idfes a f , f e .^ Pnftend totnm tri4^fUmf f e g 

0£fufUtm erit triattguU I ^. ^pderat^ cMcndendim^. 

'CoroUariuin Ptimum • 
J?r^> tridngulum f c g^/iffum Jtfrimis tribus fdnge^tith^^fim 
tnj^lum oftettdetmr eodgfn^nt^o eti^m tri^nffdi n g p . &ftti^tt^ 
teMfemferquddrtffltim erit dtioruntfftt^ultr^dttguw^^^^ ^ 

qudfo/iifJumCdu^Avtrit^ufdUdtt^en^ 

Coi^QlIarium fecttiuiuin« 

Mdni/eftum etidm eSf tridnguUtm f e gfuh t^Ujg^ftfiftts Mt|u 
tentum^y dufarefUtfqjUdmdmi^uft^ottfiUn^o^ m^to^^Jt^ct zfi^ 
qnidemtridnguli^miz^dimidium eft dutnrtpt^jiittotdtridstfftU-^ 
Tum e b a 9 e D c . Ergo eriffU^qu^ dimidium trilinei mioa^ 

Sincfeqftitnrqttodfp^ilefttinitifffird^ 
fgurdmrUiUinerdm infcrtl^ere fer contit^ttnm jdnCtnm tdngett- ^J^ 
tittpt; qud quidemfigurd^mfcrtftd ieficidd. t^figummi^d^ defc^ oimt^ 
^umittoriqt(dmfitqndli6etddtdmdfftitttdd^. 

Sip^raboladuascangeQtes h^buerit a4 bafim r.djebde per 



ia De Dimedfidne ParaboTas 

turi&hocfiatquotiefcunq^libuerit: figura atangentibus 
cumfepta » fi ex vertice parabol^ fufpendatur ^pofita diametzo 
adhorizoQtempcrpendiculari^ aequiponderabit- 

tbc, cuiiisdid^ 

mfterh^j&du^ 

tdngi^tes dd bd- 

fimfint ae,cej 

per vcrtieem vc^ 

rohtdngensfiti 

b g . DeindCy de^ 

mijfrs (vt in prd^ 

eedenti Lemmd-^ 

te) didmetris fh, 

g i-, fcr yerticcs 

fwtienum^hhy 

D i c , tangentes dticdntnr 1 m , n o . Iterttmqne per vertices r/- 

lifndrnm qnntnor fMionum tdngentes ddcantnr p q, r f , t u,xzi 

&fiefemftr donec libuerit: BicofigntamifinefotiusAnasfigH- 

Tds reCiilineds i tdngentibus p q r 1 f p , t u x 2 g t;circumfeftas , 

€Xfunifo h fquifonderdte : fiatutdfrius didmetro b d 4^ hori^ 

ZMrtemferfendiculdri « 
-P^ndturitdquehddidmeterfdraboia ad horizontem ferfen^ 

dicularis ; &reCtam f g , (quamcunq; inelindtionemfortiatur) 
eoncifidmus ejfe libram^ euius fulcrumfit in b . 

^bniamigitur afflicata^^h bifdriamfecdtttr adidmetro^h 




mttdtts 
Idres 



^dm.i 



infunCto y ;funtq a b , Im yforalleU , erit etidm 1 mfeSta bifd^ 
fidm inh;c" ideo duorum tridn^ulorum l f m » n g 6, centragtd^ 
isfuntin f h , g i ;funtqi fn, g i adhorizentemferfendicM^ 

yide^oaffenfdeentralit^terunt didtd triangula ad iibrdm 

f 2 . exfunStis i& g, Aequifonderdbuntque ex difiantqs aqnd^ 
Iwus b f t b g • Cum iffd quoque tridngula fint aqudUd ; nen^e 
fuko&ufU eiufdem trianguU f e g . Eademfrorfus ratione fofi^ 
t^ri 1 tap^o tfidn^Up 1 q , r m idffenfd erunt exfunSis 

&m;fqui- 



' Pf^l€maPrimutn# %i 

tS^m^ ^mf^ndcrdtuHt^; exfunito h » cjr ideo Affenfd efunt ex 
fm»&oi. ( qudndotpiidem flmnfuffenfionis {h ferfendicuLote 
e/t ad hmzontem .) Duo vero tridnguld t n u , x o z , frddiSfis 
dtqualid(^cumJintJinguUfubo&ufldfqudlium\itti^Xi%p -)fM i^m^ii, 
eier^buntfmuldnihoexfunStoo^^ Ergo qudtuor fmul frudiCfd 
drianguU dquifonderdbuntcxfunSo b , nemfe medio totit$s li^ 
br4t£g: EQdemmodocondudemusreliqudtridnguUiquotcun^ 
qttejint , ixfuncto b dquiponderdre • Vmuersd ergofgurd d tdn 
^emtihuscir/iumftftdexfun&ohdquifonderdbit. ^od(^ci 

Cororiarium L 
HincfroCorottdriodnimdduertemuscentrttmgrduitdtisfrf^ * 
diSiffigurf , i tdngentibus comftdhenff , effe in didn^etro fdrd^ 
bold « Cum emmpgurdfrfdiiid fquifonderet exfunih b > erit 
ccntrumgrduitdtis iUius in Uned quf exfundo b ducifurferfen ' 
cii^uldrisddborizontem ; qudfrofter eritin b d didmetro fdrd^ 
bodf^ 

CoroQaribn IL 
CoUigenfUsetidm centrumgrduitdtis omnium triGneoru mixr 
iorum , quffub linedfardbolicdd, b c , &fub omnibus tdngenti- 
bus^^^^c^xUyxxTCy^omfrfbenduiMn^yfemf^ 

bolf exifiere ^ Patcbit dutfm hoc modo ^Centrum trafetq^z, f g c 5 x sfftmt 
efi in diamctroicentrum etidmfdrahotf eft in didmeno; ergo cen '^««¥«* 
ttumrtUqudfigurfmixiferitindidmetro ^Siergocentrumbu^ ^.fecun^ 
iufdf^difi^d eS indUmetro; centrum etidmfigurf d tjmgenti- di ctu/dL 
buscircunfeftfdemonJlrdtumeftejfeyadidmetrOyfrofteredcU^ ^; f"'^ 
tnfim omniumfimuUrilituwMmyquf cAntineturfubtdngentibus ^^^^ 
{jt linedfdrdboticd^ erii in tUdmetrofer S .frim. Aequifond^ 



j^emmd/^ V^ 

•; Si parabokchiastangeiites habuerit a&emmpcr verticem,, 

alteram vero ad b^iiim »& ex akera parte bafis habeat paralle- 

lain diametso ; figura fiibtribus pra^u^ redi$ lineis y & aam 

Yft9^M)kXw^ pundo [tangesb- 

^ " ; tisver* 



• \ 



• . / 



i-is^dK»iIfc\m <|^ fiodiuicyiifriyc|>ar&adreKquamii3tfi^M 
tcm«0rii)Mat^>4upbfitiIln&c|uea^ diiinj^trQ ter- 

Bjl^of^fl^dbdd a b c^ ^M9.singm^M4hdfihJk<>AyWvertK 

/»&, 4M1// i^^ftpUfitftliqu^e^ g% :Bi^figurM'z b c t^ fAcftr" 
r^didmetro ddh4rriz>.f»f€wti^Ml^i)itfHi^mkr^texPMM6fpc. 

enimdiametr., X^ 

fdrdboU efe 

horizonn fer^ 

fen^icitUr^m 

(hocdnfemmt^ 

d^sefcr inseU 

ligendmmeB) 

qnd.mcfin4j; tS 

dem inclindsio 

nem forfidSttf 

librd gf. Et 

dniht tdngen^ 

te^2L d {qttdi am 

nituttTdnJihitfer^^ vtit^i ^ttdentfkt) inteiligdtnr g f:lHrtt 

ep^cuinsfHlcrHmeJl^iexqudftndent ^vmt fdrte tridngu- 

lumzQcididlterd^er^^yJlgurd^miixtdzbcfi. J^d quUhn, 

fgurd.f interfe n$n dequifpnderdnt^ fondmuj titerdm iffttri 

ffdefandtrdre , Efie igitnr ; &fr4efonderetfrimifz b c f c, tdmm 

t0 excejfu qud^tum eBffdttumV. 

InfcrihdturUtrdiffdmdUdfigur^itdngentihHc hilmapp 
q f c h , termintttd^ itd vt retiqudefertiunculdejhh tSffis tdn^ 
gentibus & curHdfdrdholicd contentde ^fimul minores fintffd^ 
tiok (quodfierifofie conftdt ex CoroUario Secundo Lemmdtis 
Tertq.) PrdefonderdhitigitHrddlntcfigurdftAtdngentiyus 
comfrdeltenfd ; qudndoquidemfdrsdUdtdndnor efiexcefin K; 
& in eodemfunao bfonderdtfimulcumtotdm^gnitudiuei tdtm 
tmm4bldtde > quim ntitts^ cesttfumgrdmtmU ^indidmetro^ 

vtoBcti^ 




Probtema Plrimum . irj 

%M^tnMmMS4dCoYoUmMmStcMndmmL€mm4tit Sg'^^- 

jiccjfidimr iJh» g r^maf^^f4rs^iMus g a ; dMCHiwr^; c r . Sm^ ^^ ^ 
m4i0r (fi4m c 1 dMpj reiifMei gs& tx^fMnCii 1 centr^liter fuf. ^^f^ 
f$^r$m triiptodUhft triM^pdMmk^hens 'uertictm in e fnnBi , tuxA fM* 
0^ P4jim4nreit4 g a »(^4^ ^dMriMmemrtifafonitnr ^ ^^^fj^x 

I4mjic idif$iri4ngMl4Ttx^ Vkfty 4d tri^ngnlnm ^df.Jknt ii[^i^ 
Vidno4dJs&'4dtri4nguLchdvt2.4d4.(jr4d4eq$Mlez^t. 
vt2^4d4*ergo4dtriangulMmz,tc^eTtmtvt 2.4d i.^nemfevt ^Xp^ 
\^4dc%yMoc<^vtlt^^tbrecifroce. Stg^tin &ir4 Ib duo Um. g^ 
fraeM&^Pi^ifj^uUzi^ii^yiii^dequ^on^ Kttek 

fun&oti 

: Sum4tur ii^irum^ { qu4n4 fdrstotius g r ^ (^ iung4tuP tL CS 
tmi%ttj/itqu4^t4f4tsifcius o^ty i^l ifjius ]t dh y fWgr* 
0&4if4f4rs fMUf t <1 f k j^$tkfnifitft4eptfde trO^ gr « y^&utro 
i^otum z ^ K t, u fi: ^ Md^um^m h e i tfi^4M4f4rsifjiurasLy ^^ j^ 
oNmtifU4t$orj^n^.friimgodJlihtiA^m^^^^ mat€ i^ 

I$a^4li4/utl^i0it*^)^tri4t^j^ Mdt;fiutvt2^ 

4d 4 :. &frofttt€aet.i4m4dtriangulum orejwuntvt^^ dd4§ 
udiffum vero fr e erunt vt2.4dj,n€mfe vtlc^dch^ Equi^ 
fondcr4nt igitur exj^n&Oi thinii tti4ngMf^{t e , c^ inde fU4^ 
tuorfraedi&a triangulah z i > I x m ;n flo , p tq .. Bodem fror^ 
fus v^do^ifuh qu4tfor his tfi.4ftg*4li4fu^i^tbi rejiduis foftlfk' 
culis triang. exordi)^edtfcrift4sofitn4tn^4e^[uifonderdr< ik 
eodemfunCio e, cum quod4m tri4ng. cuiMs vertexjit e, hafis ve^ 
to contineat $ . qu4rjffius g {drc^Stdin, nojltr.o caftty cim. demom 
firatumfiifrim4 duo^iangula. z ex, af c> aaquifondtrdrt tridm 
gulo,^tc.^eliqU4ittmqitMtuor4t^gfdidyq^ U 

4t^uffondcrart trianp^ iV e ^ ikequifojfdtr^tii. tot^Jrmulfign: 
tA^iiTfraedi^istHM^tif^omf^jtJri^ f , txfunfioc^ 

Stddemon3ratMmfiiitj t4ndemfiguramfr4tfonder4rt trijmgu^. 
/^ a e g > necejfe igitur eH. vt trHtfipiMn a c g minus /it tri4ngm 
l&^t \^H^^}ul4fdrtU£^tiH. mfo^le^. 
^nKTO fontiimts ,fr4tsfotidejt4rfi tviunguhfmzt^figurAtz h c^ 
§t< MStA&fito^ceffitsxiimfnuffniiky^ Actifiit^ 

i«r$ j$C^«sr/f^|^«r^^ i»it»fimpnr%it%. 



^4 De Dimeufione Parabolf 

fffblU addiquotlmMgulmm g f e, qHddmmusJlt/^dth Ic . Ttmc emm 

^^^l^ mangulMm a it Adhuc fTdefpndetdbit figurAe a b c f e • Sed<0^ 

' demmcdOyVtfufradcmimftrAbimustrUngulumiffkm^icaefM 

jbonderAreMicuifgurde re&iUnede in/criptde mtrdfgterdm mix 

tam^\>cit% JSfeceffeergpiterum^ritvtinfcriptdjigtnrd reSfp- 

Unedtndiorft qudmfgttrd mixtd abcftiCui ifjd infirihitur;^. 

fuototo^ ^odeftdtfurdumdt^. 

jteqmfonderdtergo exfuncto c^tmitfef/dfgurd abcf^^ 
fub cuKttdfdrdboUcdy duakufq. tdngentitMs^t^iinedifftUame^ 
SrofdrdUeldcontinttur. ^upddrc. 

^odafumfiumeftifdoftendemuj^ Nemft fdngentem B, d 
trdnfire ferfunctmm e .* hoc eJi , itdfecdre rectinn f^ > vt pars 
{cs^ufUtJltreUqudecg.SecttemmadfJttgmTrectdmfgx/f'- 
eunqiinCy Idm\cmmfdrdlleidfnfz^^hdyK5'^qttdlts2iCfidi 
eruttfdqudlesttidm^c^thy StddqudtesfunffhyhCyfrgofe 
dupUeififftiscg. Idf0^dfr4nsitferiUmdcfmit€fmm ^ qmod 
od imitio dixordmus . 



m 

Tr<fofit$o Prms. 

VABLIBET parabolafefqintenia eft siangiiE ean- 
dcmripfibafinit&fandcmalrinidmfinliabcnns^ 



(V 




Efto pand>oIa A B C, adus 
«[iameterBD,Hmgaanirq.AB» j| i 
B C, Dico parabolam ieiqui- "^ 
terdam eflfe oianKuli ABC, ean 
ilemcumipfabamiv&candeDi a| 
^dnidinem habend^ 

Ducanmr tai^entes A E,CF 
ad bafim : F H ver6 per yerticcm Bj & A H fit ipfi diametrp pa- 

rallela* concipiamufii. paraboke diametrum eredam effe ad ho 
fizontem. IamieaaH£inI,itavt£IdupiafitipfiusIH,eric 

fdaoguhimHAfceatraUtbrappei^ (M^^ 

. -_- .- ^ 



nim centrum grauitatis in|^re&a qux exl ducitur parallela ad 
H A , & propterea ad horizontem perpendiculari • ) Erit infu^- 
pcr figura mixta A B C JF £ centraliter appenfa ad pun(5lum B . 
(quandoquidemhabetcennrumgrauitacisindiametro BD;id rii^ c« 
horizontem perpendiculari .J Sed vniuerfa maghitudo, com- roii.u 
poiita ex dido triangulo H A £ , di<5hiq; figura mixta,aKiuipon- •''•♦• 
derat ex pundo £s erit crgo triangiilum H A E ad Hguram mix- Ummi 
tam ABCFE , vt reciproce B E ad E I , nem.pe vt 3. ad 2. Pro- f^^^^' 
ptercanrapeziumAEFCfextuplumdiditrianguIi, eritadfigu- onende*^ 
rammixtam ABCFE> vt i8*ad 2. &per conuerfionem ratio- '«v!/^ 
nis , zd parabolam crit vt 1 8, ad 1 6. Qualium ergo partium pa- 
rabola cft 1 5, eanim trapezium AEFC eft x 8. & triangulum A 
BC r 2 . Qu^eparabola ad trian|ulu«i ABC erit vt 16, ad 1 2 ^^*'^^*- 
nempefefquittrtia. QuodcratQUcndehdum. 

^$dtTdfexium a e icfextuflumjit tridnguli h a e , fdtef . 

NamfdtAlUhffdmmu\A^duflumeHtridnguli)\2Lh (jrfrofte'^ 
red quddruflum triduguli h a e • eryi trafexium a e b d trif^ 

lum erit trtdnguli h a e . totttm vero trdfezium a e fc . fextu^ , ^ 

flMmdi£titrianguli hae. ^uod^c. 

JljCum dutem trafezium a e {cfextufltimjit tridnguU h a C,^// 

fhituflumetidm trianguli e a b ,• c^ ideo triflum duorum e a b » 
b c f , Nemfc vt 18. ddif. Per conuerfionem vero rdtionis , erh 
ddtridngulum^hcvtig.ddj2. Sluode^c. * 

Lemmd VI. 
Si dua? parabola? vtraque duas tangentes ad bafim habuerit ; 
crurtc inter fe trilinea mixta fub tangenribus , & curuis parabo* 
iiciscontenta,vtfuntipfa triarigula fub tangcntibus compise* 
henfa» 

Sint dudfdrdboU a b c , d e f . qudrttm vtrdque duas tdngero 
, tes ddbdfm habedt ag, ^cfrioris^ c^ dh ^ fh ,foff(riorisfd* 
fdbold. Dico triRneummixtum ^hc^, ddtrilineum mixtum 
defh, efevttridngulum agc, dd tridngulum dhL 

^ i> Sienim 



<^ 



_ ^^^ • . 

7j6 DdXittenfioiieParabote 




lm.%^ 



•^« 









SienimMneJHtd: hdhehitsitefMmextnlintis^fMtd abc^ 
^dteUqntnn , miuoTemTAtitnemqnhnpndngnlmm a g c> Md^6k£. 
ESo^dtinm k excejfusyqnemlincnm abcg^ tnnins ejl qtthm 
vt Jit inTdtiene tridngulirnm ^ 

DttcdtttTfeTverticem b, tdngens i 1 1 demiffifq; exfttmlfis i^ 
tjr \ylineisdidmettofdrditcUs(qnddidmetTifemifdTdboUr$nm 
dTunt) dn^dntuT tdngentes d m > n p «- c^ exfnnEtis o ; m ; n>p> 
dendttdntttTdUddidmetrivt/kfTd \dncdntt^fi dlid tdngtntei: 
£t hocfemfeTJidty quoufq; TctiqUfJimulomnesfOTiiuucuUy qmn 
fuhtdngcntibnSy(^ct(TUdfdrdbjoUcjitontinentuT; minoresjint 
ffdtidV « TnncjtJunincTjdfgtard tdngentibuscircnmfcftdy&in 
tTilineo mixtp JSihcg infcTiftdyhdtehitddhujCddtTitinenm dc 
fh ytdtiottemmdioTemiqudtTidng. zgCyddtTidngulum dhf. 

InJeTihdtttridm etidm in dltcTO tTiUnco mixto d e f h « fgnrd 
totidcm IdtcTum ; duciis nifwum tdngcntihus totiesy quotics 
dttHdfueritU infTioTitriUnco • 

^upnidmvcTee^ yVt tridngulum igl ddtTidnguhtm qrh» 
itd duo fimul tridnguld o i m > n 1 p^ dd duofmut uq f > trz . 
(funtenimfdTtesctnnfdritcrmtiltiflicihujineddem rdtionc.) 
Etvtduo fmultTidngutd oim, nlp, ddduo UQfytrz,itd 
qudtnoT tTidngnldqndJkntinJrdfttnitd o> m , n , p ^ddqndtuer 
tUdy qudfuntfihfunliis u, U U z ; oh cdndcm cdtfdrn (^e^Tunt 
dtidmtmmidAntd€€dentidJimtdfncmfcJi ^ttrd infcTtftd infrit^ 

Titri* 



Plt>bieiiui Primuin# vj 

witrUmi$mixto 4d4immac$nftqM€ntUfimid(nemft ^dfign^ 
TMt mfcfift4m mfaSmofi triKnea mixto) vt vnum ddvnnm ; 
memf€vt igl*^ qhr« Sine fnmftis eernm quadmflis ^ vi 
a g c i^ d h f . SedeMdeminfctiptdfiffitAh^ebdt dd trilineum 
defh snsmemrMiineme^mfittridnguU agc dd dhf. Mi^ 
jfus erg$ erittfHineummixium d e f h > qukmfgmdfibi infcri^ x^^mm 
ftd:t0tumfudfdrte. ^uad efi imfo^iile . Trilined ergo fuh 
idngentibus , ^ curuipdrdbolicd cosnfrdhenfd jfrnt interfe vt 
nids^uUfukqfdemidngentibus&hdfihuscontenid . 

Profofifh IL 

PAraboIa refqiutertia eft trianguli eandem ipfi bafiau& eaa^ 
4ctn akinidifian habf nri.v» 




SitparabolaABC» 
cuius diameter BD; 
&iitinfcripum triao 
gulumABC* Dico 
paraboiam feiquiter- 
tiam eife trianguli A* 
BC> 

Ducantur dua? tan^ 
gentes ad bafim» qux 
fintAE,CE.&FG- 
tagat per verticem B 
Demiflis deinde F I , 

GH diamctro parallelis,vt fintdiamrtri pottipnmB AlB>fiHCi 

dueanturperI,&HtangentesLM,NO. . 

EritergoperLemmapnEcedens, trilineuai AfiCE, adtii- 

lineum AIBF , vt eft triangulum A E C , ad ABF . fiue ad FBE . 
Idem ver6 trilineum ABCE ad aliud triiinenm BJH C G, crit vt 

Idemtrianguium AECadtriangalum BGCrfaoceftad fiG£, 
Coauindim ergo , erittrilineutn ABCE id duo trilioea AIBF, 
BHCG I vt triangolum A£C ad triangtflum FEG^ oempe vt 4, 

D % advnum 



^•Lmi 



M.|Mf 






* - 



* '. " 



%i 



De Dimenfiohe Parabolae 



ad vniim , & diuidendo , erit trianguluni F E G ad duo trilinAi 
AIBF , BHCG , vt 3, ad vnum . Trapeziumautem AFCG , ad 
cadem trilinea trit vt ^.ad vnum; & per conuerfionem rationis^ 
ad parabola erit vt 9. ad 8. & ad triangulum A B C , vt 9. ad 6; 
Quariumergopar tiumparabola eff 8,TaIium triangulum ABC 
eft 6. Conftat ergo parabolam infcripti fibi trianguli fefquiter- 
tiamcflTe. Quoderat&c. 

Lemmd FII. 

SiinparabolainfcribaturtrianguIum:eandem haben$ ctira 
parabola bafim, ean^epiq; altitudinem. Infcribantur etiam 
pariter & in reliquis portlonibus diio alia trianguia .* Erit trian- 
gulumprimoinfcriptum ^ o^uplum alterutri pofteriib infcripti 
trianguli . 

Demonftrdtur hoc Lemmd ^A Archimede Prof. 2 /• Dt ^tfd^ 
drdtMrdfdrdbold. 

Ltmmd VIII. 
Si in parabola euidenter infcribatur figura ex triangulis eon- 
ftans . Tam bina ipfius triangula (fi prout fibi mutuo refpondent 
ita fumantur ) quam ctiam tota infcripta figm a, a?quiponderabic 
cxpundomedio bafisipfiusparabolar» ' 
. Eftofdrdhold a 
b c , cuiur didmt^ 
terftt b d ; c^ in^ 
aerfd ftdtudturft^ 
gurdy itd vt did" 
meterdd hori\on^ 
tem ftt perpendi^ 
culdris . Seffdde^ • 
indevtrdqi ad,dc 
bifdridm in e f ; itt - 
• Tumquefeitispdr^ 
^ihush^^kridin^gf 

hyiA.&c.Ducdntttr^m^ttifho/iip.fc^^it.^c.TdrdU^^ 
Itrtddi^mttrMm. * 

Infcri* 




Problema Primum • ^p 

Inflribdturqtie invdrabolkfigura ainnobpqrc*( qud di^ 

l €$tHreuideuieriufcr$bi.) Dic0 triauguU qua figuram iufcrip^ 

:: Aiw compouuMj ftbina^ (jrfroutfibi mutuo refpondent itafuml^ 

tur , dquifonderare exfunSio d . Prpterea vniuerfamfiguram 

r infcriftAm , ex iffis triangulis comfofitam , db eodem funCto d , 

fquifonderare. 

Sumantur emm Juo triangulafibi mutuo reffondentia yfuta , ^ 
nob, h p q y qua interfe aqua/ia trunt; cum triangula anb, lam. j. 
b cffuboifupla .fint eiufdem triauguli a b c •• iffa vero n o b , 
b p (\,fubo6fuplafint aqualium triangulorum a n b , b q c. Ha^ 
bebunt infufer ccntragrauitatis in re^is o f , p t , qua quJdem i j frimi 
ab anguUs o , p , ducuntur adfunffa media bafium , n b , b q , ^»'>* • 
Cum vero o s h , p t i reHf ad horizontem fofitffint perpendi^ ^* 
culares , erunt fradiCfa triaHgula n o b , b p q ctntraliter affen 
fa exfunCtis h , c^ i . ^amobrem ab fqualibus diftantijs h d , ^iciufd. 
di, fquifonderabunt.Etfiedereliquisfigurftriangulis.^od 8. frimi 
rratfrimofrofofitum. ^'^J^* 

Tigura autem vniuerfa euidenter infcrifta eomfonitur exfor^ 
tibus aquifonderantibus afunCto d ,- quare etiam iffa ex d fun^ 
ito aquifonderabit. ^od erat ofiendendunh(jrc. 

Lemma IX. 

Poiitisijfdem. Slaparabola dematut vuiuerfa figura eui-* 
denterinfcripta, etiam omnia fegmenta parabolica, qusecir- 
cumrelinquunrur , ex pundo D . aequiponderabunt. 

Sefetita enim eademfigs&a demonftratum eftfiguram infcri^ 
ftam aquifondcrare exfunSto d . tirgofigura infcrifta ccntru 
grauitatis habet inftrfendiculo hori^ontaU d b . (fer ^f^ffo^ 
fitionemjfed etiamfarabola centrum grauitatis habet i» diame* 
trodhy {fer 4fecundifquifonderantium) ergo ccntrumom^ 
nium reliquorumfegmentorum erit in diametro d b . Sitiare ex 
funCto d fquifvn£erabunt . fer j . fuffofitionem . SlHfd (^e. 

Cotollarium; 
Conjiat ctidm todcmfrorfus argumtnrOy rtliquumfigHracuim 

den^ 



50 De Dimenfione Pmbolse 

dcMirh$fcriftd , deira&^fruis mMguh a b c ^ ^LqmifomderMrt 
€XfumS§ d • ItemrcUqMMmfnrdhU^ deti^u eruu^mU a b c» 
f^mipimladrt ex d^ 

Si ex parabola auferatur dimidium trianguli iafcripdttoca rc- 
liqaa figura mixta asquiponderabit eac pufida baiis reliqui trian 
guli » in quo fie ea diuidicur > vt pars ad curuam terminata qua* 
dnf^ iit iliius » quae terminatur ad diamamm j 

Efio fdrdBoU a ^ 

bc iHMcrfd; eiufqi 

dutmtter bditdfiA 

tuMtrvt ddhprizS 

temfitfprfendicu^ 

Urit ; Detrdifoq; 

fetmttidngulo in- 

fcrifto d bc;feee^ , 

tur ad tdfis reliqui 

femitridnguli , in 

qMinq;fdrtes fqud 

les ; quarum vndfit d e . Hieo huiufmodifigttrdm ex funffe c 
fmffet^dm^ dquifotoderdre * 

Slifienimdquifonderet; Cumre&d 2Ldfit lihrdy cttiMsfidcru 

efiin ty(^ tttdgnitttdo a fb ^c^confidtuexdMdbusfertionibus 

^^^' fdTdkoUcis ^dffeufdfit ddfunCium dfecstndtimcentrMmgrdui 

'^ Odtis iffitts : ReliquMm dMt^m tridsi^MlMm zh d. dlterd tndgsusn 

do dffenfdfitddfuniium\i (fumftjt d hsertidfdrte rnius da^ 

Alterd ex hisdudiusmdgnitMdinit.frt^onderdre nec^e erit . 

PntMtttsfrimbfrufonderdredttdsfmiones afb>bgci <^ 
fit txceffusquofrttfotUerdnt , uqudlisffatio K . 

InfcribdtMrttMenterinsrududsfortionesfMrdbolicdsfigurM 
multiUterd , itdvt omnidfimulfegmentdfdrdtoUcd circumreli^ 
&dminordfintffdtio K. TuneenimfrAfoitdetdbit ddhuc fign 
MnfcriftdmnkiUderd ^ii\\txsi^tkc)i^9k^ 




t 






Problema Primum^ 5 1 

^Jc^ifidHtr d o qndrt4f^s mius 6hi& duSti a O » Mmfi^ 
tMmifiMguUtm a b o » itquifonderabitfihiiffiixftmB^ k ifid 
€tiMmqM§dc$mqi4iUMdtridngnlHm hdhens vcfticcm im ZpC^idm 
JimintiCld dh ^fiti iffi^tqMifondcrahitJtxfunSlo codem h« 

Idmfic; ^l^utUMmfMrtinm^di effisy ali efis>& dteH 
j. £rgo dcMd eh^eritvts.ddj. CMmdMtem denmn^dtMm Lm.u 
fitdMOtridnguld a fb » b g c » Mqnifondefdre exfmn£h d t tri^ 
dngnlMmveri ho z^exfnnffo hs&cnmdMofrMdiSdmdngM* 
Itifint ddtotMmtridngMlMm ihd-vt dm ad^- icrtent eMdemdd,fx Umi 
tridngMlMm a b o ^vt 2. Mdj^ ; nemfe vthtddcd recifroce . 7« 
^M/imotremdMoittdtridngMld a f b , b g c ^sMmtriMgMk abo^ ^^ ^^ 
MqMifonderabMntfMffenfd exfMmSfo e • imtmd^ 

iMmdtMr deinde d p qMortdfars iffiMs d o i dMCdtfirqMc a p • 
Jdm ; ^$dd dMO triangnld f 1 b> b qi g fqMtfonderant ex d Lm;i^ 
itemqMcdMO a i f , g n c > dqMifonderant db eodem fMnSto d ; 
Qmnid fimnl frJttHifM ^dtnor triangtda fqnifondetrabtin» ex ^ 
fMn£foA; jSlj^tMordMtemfr-fdiSfdtriangtrUadmangMiii a ft> 
futtt vt dMO ad^. Snnt antem a fb , a o d , fMbquadrufla eiuf ^* ^* 
demtrianguli abd &froftereaadtriangulum xoi^jMttntvt ^* 
2. ad 3 . nemfi vthc ad edj recifroe} . Aequifonderas^t er^ ^i «^^* 
goqudtuorillatriangutacumtriangulo aop, exfUMSfo e» £r ^^^^ 
go vniuerfafimulfigura euidenter infcrifta aiflbmgncba 
aquifonderat triangulo a b p . SedeademLfrffottderabdt triatt^ 
gulo a b d, Minusergo eH triangulum a b d quam triangttlum 
a b p :totumfuafartt.: quod efi imfojftitiia ^ 

Ponamus 1 ieindcfrffondcrare triangtdum> ^hdi& fit excef^ 
fus quo fTffonderat aqtuUsffatie K ^ 

AcctfiatMr a o d qMOtta fdrs ttuku triangtt£ ab d; itertmt 
qttefMmaturz, y^A^qMoridfd^s triattgttlk aad lethacfemferfia^^ 
domcvematMrddaUqMOdtriattgMlumyfuta apd yquodmittMs 
fit qttam^atitmt K. 7unc ettim neliqUMttz a b p adhttc fnffon^ 
dtrabitduahusfo^tionibusfarabdiicis afb,. bgc Sedidem 
triangtdttm itfetodetstr. (eodemfTixrfut ttta^ vtft^rd ) fquifott* 
derareaticuifigtti^imt^fatiahalicasfm^netij^rtfttf: ttecef 
fe igitttr erit qtiod fmi im es farakidiaf mmaresfitU qwvt^ fik'^^ 

^illd 



ftdtii. 



3 X De. Dimehfioiie P^abolg 

illafibiinfcrifta ; t^tumfuAfaru . ^Md eJHmfojjU?iU .jiequi^ 
fondnant trgo farabola inuerfa ( de mfto femitrtangulo infcrif- 
$0 ) €Xfun&0 quod di^um efi « £luod eroi oBendend. ^c. 

CoroIIarium 
HincinfirrefoftmHs ^ quhdji ex funSo e, relta ducetur 
diametro aquidiltans, centrumfraediStaejigurae erit inprodu^ 
£ta. SiquidemfguraiXfunilo e aequifondtrat j &linea ex 
e duifa aequidijians diametro , c'B ad horizontem ferfendicu^ 
laris. Poffetetiam demonftrariy nijiextra rem tjfet^ centrum 
fraediiiaefgurae diStamfarallelaitafecarey vtfors quae $cr- 
minaturadcuruamjit ad reliqua?nvt 1 1. ad u. 

Profofttio I IL 

PAraboIa fefquitertia eft trianguli eandem iibi bafioH & can 
dem alcicudinem habentis • 

Eftoparabola ABCexqua - 
demptum (it dimidium trianguli 
infcripti : Sumptaq; DH , qu« fit 
tertia pars totius D A & D£ quia 
ta pars eiufdem ; (i parabola hu- 
lufinodi ftatuatur inuerfa » ita vt 
diameter fit horizonti pcrpendi 
cularis , oquiponderabit figura 

expundoE, Sed triangulum ABD appcnfum cft fecundum 
eentrum grauitatis ad pun(5lum H libra? HD • Du^ autem pa* 
rabolicas portiones refiduas appenfo? funt fecundum centrum 
grauitatis ad pundum D ; Ergo triangulum ABD, ad duas re- 
liquas portiones erit vt DE ad EH, r eciproce,nempc vt 3. ad a: 
Sumptis autem antecedentiu duplis erit totum infcriptum trian- 
gulum ad reliquas portiones vt tf. ad 2. Conucrtendo igitur,& 
€omponendo,eritipfaparaboIaadinfcriptum fibi triangulum 
. vt 8| ad ^t Ncmpe fefquitertia . Quod &c. 

' Libct 




7rcfldema Primuiii 4 ^^ 

tHef bie dttiMnpdfe. LemmdJjte* Vder^^Bn fdMtenmi. 
de^ ditierftfiiieftmtit* lletbdniedfrineif^s ; jffe emm veitne 
Ftef^JlrieneitU , f «i dnte demenflrMerm eentynm grdnitdtie 
^^/^bdr^.NttJUitemfimiirdtieneydeinfrdeedentibne.d^ 
mettitrMbimm ^ Lemmn y&effdm r^er^Mtnelnfienem, * " 

m 

Lemmd Xi, 

X>mnisTeinii>araboIa«quiponderatexpunaoba(ts, tn quo 
fic ea diuidinir vt par5 adcunuoitenninata fitad rdiquamvr 
•^ttintfueadtriar:^ ^ 

^tfamifdrnbeU zhcy-eitine 
eUAHtetet ^JkuwjitnrndikerixMt-^ 
,*emferf*ndienLms t StUadpnde 
ac ^in f, itdvt tidd f a » // 
vtx.dds.vitvt ij.ddfi.liieef* 
ptrmnexfnn&i (fitf^eifd;»fgMi^ 
fenderdee k 

teeetitriteimm ac btfariimin 
Ay^demifsil de fdr^UteU did~ 
.metre ,eritiffd dc, didmeterfd' 
.tdbelf bec . Snmdtitridm ai /ir- . 
tidfdrrtetins ac. \^nmigitnr fdrtinm ac ejfi^.tdiii 
ad ^//^.af^.t^ ai jf, £rgod{tresy& fi vim. /^/ 
^dnendifnifMdirdtexfnnae fiCnm id ftUbrdqndddm 
emnsfitlernm efl f, d-ddfunanm i dffenfnmfit tridnenlnm ,^ tu^ 
a bci exfnnaevefh AdffenfdfitfdrdbtU h c c ^terdex bis i>LS*. 
fgnrtsfrefonderdbit. PtndnmsfrimhfrafonderdrefdrdbeUm 
iiecyfita; exeefns qne ftdfenderdt dqudUsffdtio K. 

Jiferibdtnr enidentW intrd fdrdbotdm b ec /^iwrf r*/?//A 
"'^yt^ytomnesfimntrefidndfirtinncnUqnibnspdrdboUex^ 
f^dtetnfcriftdmfibifignrdmymineresfintffdtioK. Mdnifcflm 
'fl^^»tdtnfcriftddnidemeef^nedJulbncfrafonderdhitJridn. 

^f<^»>M/jrahc qitdrtdfdrstetimtridngnUtihc^ €nmdn^ 
M» d ^flf^d^itu/itemferfenditnldris.^tnangnUm b c* 




^ DeDimenfiQaeParabote 

3 i frimi AMf€4iH9»mngr4Mit4Usim TtQd g e $ tfitdi&mm ttidi^bnm 



ff^if^^ff* 4fpt^im4id d . HriMgMhmvath b h c ^pcnfim ddfum3< 

i if$49thfMidfm ai t€ttutfmstfit0tmsd.c^ifj4niMf9 zh fn^ 
ffMdicMlmsMhmT^eme^nfiitMtdtfi^ ^^ttttti^Msem h 
c^CJifl^hc^fivtvmMmAd^^tritidcm }Bit c dd \xh c ,w vmm 
^* ^^ 4di. nemfe recifroce vt ifjid f d, AeqMifondcr4mt ergo exfum 
ifo f, tridngnU b e c, t^li hc* 

Sttm4tMriierMm a lc ^M4rt4f4rstri4iigtdi a.h c ; £# -^m^ttiM 

dno tridngMtd b Hi c ^ e n c Mqmfonder^^t ck g (yti demQmHrM 

lim.t. tMm eii) AqMifonderdbnnt etiamfMjftnfMex AJuttm4S^tmdtt§ 

JL$m.7. ^i^^^^i^t^g^td hmc ^tnCyfiMtddfri^mgtdtmbtCjfitteMdif 

' Ji4qM4le a hc , ^/ vmtm Jid^^^^ ertttti jid khc^^wvmMm 4dji 

ncmferccifroce^t i f W f d * jitqMiftttder4ttt ttgt 4xftttt0M 

^ f.dMotridngnt^ bme» tnc-^sMmtri^ttgtdt Ihc* Fifftrditrgh 

vniMerfd CMidentcr infcriftM imtf4f4tAol4m b e C mqtiiftmU^ 

r4t exfnniio f • cum triangnU 1 b c • Sedt4defMfrtfotoder4k4M^ 

triangMlo a b c • Ncccjfe igitttr tB qmd triMngnlMm abc mi^ 

nMsJittriangMlo 1 b c^ iotMmftt4f4rte y ^odiJi difttrdtm . 

PojuMMs deindcfrMfomicrjteetriMttgtdMm a b c , &fifexcef 
fMs qMOfrsfonderat 4qM4lis fftuio K. SMmMtttr ahc qtt4tttA. 
f/srs totius triangMli abc. ItcrMmfMmMtxr jxlc qM4et4f4rs 
tri4ngMhWiCy Et^fettf&fiMtydonccvcnttmfiterit^d^U^ 
qttodtrijmgMlMm ifMt4 ai c, minMsff4tio K • TMr^ctjnimtriatt^ 
gultm 1 bc Wi&jsrr frMfor^dtr/AitfMt/dHflf b e c^ Sedeodem mo-» 
doyqMefMfr4ydcmoniirabimMsdiilMmtri4MgMlMm Ibc aqMifom^ 
derdrttMidamfigMrteMidentcr infcrifta intra f4rjA»l4m. b e c • 
VndefeqMeretMriffamf4r4tolam bec tmnoramefit^UqMMfigm 
rafibi infcrifta \ totMmvidelicetfMafatte « MHfdefiabfMrdMm^ 
jitqttifottderat ergofostdf4rJk0l4 , ^^/# diSitm tH ttatfiiittt4 \ & 
^xfMm&^ifMffexfa. ^jUtdf^c^ 

CoroUanum» 
Hincfatet , ^'/^W^ cumfemifarabolaaqMifonderct exfMtiSor 
snffofi^ i^)fi4b (demittMtMrriSjtim^mti^erfomdicMiMnsJmk^cde'' 
/i« 4 . mifatritxetttrMmgraMiutixftstnfmahtia ; aliioiatim non fqmk. 



' Pn>blema Frimtim^ j^f 

f0$iir»€t ex f . Vnkmqu^mdm ttum diamcter pMtdMf d d 
bmMntem piffendicmldris cMBinftdift f cemclddemMs; fiA d^ 
Tc&dquM exf$m&0 f dncitmdidtmttc fqmdilidm y ttdtifitftt^ 
^ctttftttnfmmfdidkcU.^ 



Trif^pth IV. 



p 



Arabola fefquitertia eft tmn^ eaadcm ipfi ba&n» itnat 

demaltitudineinhabentis r 




Eftoparabola ABC, cuius diar 
meter B D , triangulum vero iiv- 
fcriptum fit ABC, Dicoparabo-' 
lam didi trianguli eife fef^uiier- 
tiam. P E^d 

Sumatur,qualiumpartitimHK > 

taDCeft24^taliumD£8.jDF^;&£)Gt t«« Bricqtteei» i 

ruiRlem ££vna»&FGtres^ Dudisver^EHtFl^GL^diac^ 
netro paraUelis , erit inEHcenmim urianguiiBD Ci inFI §f.ffiM 
centrum femtparabolft DBMC> &:inGLcentrumportionis^^/. 
BMC, ^^J^f^4 

Ponacur centnim trlanguli efle pnn^m quodcumque H ^^^./h^ 
Item centrum femiparabolar eflfe pundum quodcimquc I t»4p» ^ 
(quamquamhuiufmodipundaextrdipfas figura^ vbicunqj li» 

buericfuniantur^tameitvenmifen^^ereodemodoinferemus.^ «• f^ 
iundadeindeHI,&produ<ft^,inipfaHIeritcencram pordp- ^f^tf^, 

nis par abo&cf B M C ; quod cum fic eeiam in reAa G M prod»* 

£la > neceffario erit in communi concurfu L * ParaboU erg6 B 

M C ad triangulum D B Ccritreciproc^ vt H 1 ad I L» hoc efl^ 

n£FadFG>nempevtvnimiadj. Componendo ergo , fum 

ptifq; duplis , erit totaparabola ad tomm triangulum.vt 4 • ad }• 

Nempeiefi]uitertia. Qupderacpropofitumf&c. 

kteJAmnMt* 



*» 




n •^i 




^tUUn^&dimemqMddn^dm^dMSii^; terM Uiere, ^em^ 

f^dtMr*md9$guUm . V Mumfdhde.figm^d fqmfenderdhit ex-fm^ 

3q iertij Idierisy im queficeUmiditm vtfdu-dddmssdmtnmUsui^ 

tdfitddreliqudmvt j.dd j. 

Eftefdrdbeldibc^cMiusMdmetde^ 

d b ftdtudtuf ddhmxautem ferfertdi^ 

etdHris i cinfideretmque if/k fdsdheld ^ 

inuerfd:Tumdd dtserutrum idfis a c 

extremum ^futd ddfunSlum a , dditm 

gdSurreHd zt^didmefre uquiMftdsts^ 

& ^fH^^ didmesri quddrufld . . BucU^* 

deindesentio idtere^ c c sridnguU c a c» . 

feceturin U itdvt ci. ddit.fijtvt^ 

j. dd j.^JDice huiu/medi figurum ese: 
OSUmic f^^^^ f fquifendetdre. ^ufimidm^ 
iunmfrs enim ce er^ndtimMfUcdtmddsUdmetrum^SLti^tttdfigu^ 

Tdu «^a b C fmifdrdbeld^ Erge jffdem rdtionibtts % eodemqifre^ - 

dqu^nderdre ex £ Ssundtm idmJ^ i o3o edrstm fdrtsssm ; qud^ 

» . UumtotdtceH 24-&t\i2.& ti p^Eritqi ii edrmndemi 

.V , lyf^^& fl ^* ErgftcumfdrdhUzhc fendedtex-fMnSfeAj. 

*\ s^^ttfd^ iffMmffCutuium centrtMngrduitdtis;tridnguUm ve^ - 

r^ aec ex,fun{io.i;eritfdstdholdahc ddtridnguUtm aec: 

. • . inntrecifrocd ii dd iljnen^en/tvnumddsiifiMestt^. dd.ts;. 

^^^^fr/f/eredddtridngulum ahcut^. ddj ^ c.Eft enim a bc : 

mmni bmqMdstdfdrs iffius a c e &c. Cenfidt ergofdrdhoUmfefqtutmii^ 

/f ^,9 epinfcriffifibittidnguU. 

in fM9^ SttoddffumftumeB , nentfe rectdm c e ordindthn dfflicdri ddl 
ne quMi dfdsnetrtmt. a e > ofiendemus hoc modo ^ . 
^^ * SietdmnoneftordisuuimdffUcdtd< e ydffficeturdrdindtim 
c m ; eritq\ m a b c femifdrdhold ; & quiefunt fqudles ad> dc, i 
•fc/tfnfi- c}^ mcfectdh^/ridmcn n« Erge n!k'dfffyMitmideft>if^s^n^ 
t^f^lih;fedetidm ea oh conHructiontmfeJquitmid e3,iffists Ibt 

... ergo. 




__ Probtema Primam •■ 57 

^fiAfretpfM ^m ftfymitertUeBreli^i \n;& tcfefiHi ^,^^i„ 
seriiaifJfMf c 1 . efMedeftimfefihile . E» etum duflk , n»n du^ ti . 
temfefquitertid, ^4re»uU4Midfrfttf cc exfUHcte c *ri»- 
»Mm4fflismf»tefi4d4iMmetrittmt. SjS^&^*- 

AraboUfc£^tcraadl triangolieanuldnifEfibaiiai >«an* 
deinq; altitudincm habemis ,- 

t 

£ftoparabor«^ 

AliKsf CUlUSdlSI» 

incter,BD^ian«^ 
gulum infcripQiifr 
ABG^ Dico]» 
rabolameffe kC^ 
quitGitfani trian-^- 
guliABC. fibi. 
inicfipti^ 

Si enim ita m>n^ 
oftrneq;triangu-* 

littn A B C erit tripfum^duanim flmul reCquarum portlonum A 
E B ) B F C Sed eritveLmagisqaamtriplum^fiue minus quam 
trifdum»** 

Sit primd minturquam triplum » eruntq; du« refiquaf port&o* 
Be^ni^is^quamteniaparsuianguli ABC» Efto excefTusa?- 
qualis fpatio K » &inicribantnr intri portiones prim iim triaiK 
gukt ' A £'B , B F C ; iterumque in reliquis portiunculis quamor 
triangula A€B,EHB,BLF,FLC;deindeoao&c. &hoc 
femper donec eyceHus portiotunvfupra infaiptas euidenter B^ 
guras (k minor fpatio k • Tunc .n • erunt infcriptae fignraBodfauc 
maioresquamtertia parsttianguli' ABC« 

SumaAipiam triaagulum ABM^qnariaparsrotius trianguli 
i^C^-EtquomamABCquadruplumefftamtrianguliABM/' ^* 
q^trianguloriun AE 3>B f CfinMiI fiiinptoronHapqualccgit '* 

tnan» 




^S^ pe Dimehfione Ptrabolx 

triatigulum ABM dubbus (imul triangutis AE B, BFC ;Sr 
proptereatriangulum M B C triplum erit duorum fimui trun^ 
gulorum AEB, BFC. 

Aocipiatur ieeri^ triangutum A B Mqua{fsi|fan coeius triait • 
guli A B M . Cum ergo A B M quadruplum fic ipfiut A BN; & 
ljm.7. duo A£ B) B F C,quadrupiafintquatu6r fimul fubfequentiu 
triangulorum A G E > E H B > B I F » F L C> cumque anteceden 
f ia (int aqualia » aaqualia erunt etiam confequentia>& propceiira ' 
cum triangulum NB M triplum iit triai^uli AiBN, tr^itltim cui . 
erit idem viangulum N B M • quatuor fimul triaqgulorum A G 
iz ^^ E, E H B, BI F , F L C Et vt vnuni ad vnmnAa omnia fimul 
ad onmia # Quare totum (imul triang ulum N B CyOriplum ent 
figurarum eiliident^r idtrd pordones infcriptarum « Sed triangu 
lum A B C mimis f rat quam triplum earundem : Ergo A B C • 
minus eft quitai NBC totum fua parte • Quod eft abfurdum &c« 

Ponamus deinde triangulum A B C euc plusiquam triplum 
duai^um fimul reiiq WMnm portionum . Efto : Sc exceflui» quo eft 
maeis quam triplum , aquale fit fpatium k. . - 

Accipiatur A B M quarta pars totius trianguli A B C • Ite- 
rum fumatur A B N quartapars ipfius A B M •: £t hqc femper 
fiat donec veniatur ad aliquod triangulumtputa A B N , quod 
minus fitipatio K . Eritq: adhuc triang^lum N B C magis quam 
tiplumduarumpoctidaum» Sedektemprorfusratione,&or^ 
dine quo fupra f oflendemus triangulum N B C triplum efle cu« 
iufdam figuraeintraportiones euidetver infcriptae ; NecefTe igi- 
tor eritquodportionesipfarminoresfintquam figura^ intraip* 
ias defcript^ : Totttm fua parte . quod efl impoffibile * 

TriaDguIum ergo A B C duat utijtreliquarum portionum tri« 
plum cd iSc componendo j dc per conuerfionem rationis para^ 
bola ad fimm trianguhim erit vt 4. ad 3 * Nempe fefquitertia t 
^od erat propofiaun &c« 

Lintmd XII. 
* ' Siparabola tres tsmgenbes habaerit » duas ad bafim > tertiam 
vero per verticem : ]&it ttiangulum fub tangentibus compRB^ 
heofilav Kliquc figucci(dttmpcd paraboli^ tqplum ^ 




mestr hAitsm^ 

M€m%ter$fbo^ 
. I>ic^ tridi$gii 

tdt$getiyus com* 
frfbemfttm rtU-^ 
^^figtttf mix^ 
ttt€ abcg f/4i/ 
fidfcili€€tf4tAoU}triflttmt£k. 
. ^i€mmttM€Sttriflt$nt^grit€^^^ti^€Lm4giSyU€lminksqu^ 
fTtftttmm 

SitftitwmiuMsquhtt triflumi tritq; rtli^djgurd mixt4[ 
a b c g f > mdgis .quJtti tertUfurs trianguli f e g • Sitrttxc^ffus 
K . Ducdnturqif€ru€rti€€s dhfciffkrum fortianum tang€nt€S 
h i , I m ,' Jttrumqucfer uerticesfulfequcntiumfortioHum, tJU 
g€ttt€s4ganttir n o> p q, r s » nv. ^ bocfcmftr ; doncc cxcejfus 
fgtira€nnxtA€ a b c g f , f^fra fguram €x irianguUs conBan^ 
t€m n o pq r f t u gjl'^ minus aUquando rilinqtfatttrquamffa^ 
tiumK. Tuncenim eritadBuc fgura €x trianguUi infcriftd 
maior quam tertiaforsfrianguU f e §,♦ 

Jtccifiaturtriangulum (Qi.quartafOrt tri4nguU feg,- crit 
qu€ triangulum fci acqualcduobusfifmtltrianguUs hfijgno.^ 
(cum tdm ifta duo y quam iUudfolum y/uhquadrufU fint ciufd^Zim^ j^ 
trianguU feg)£rgotrUnguI. ie g trifUm crit duorum pmul * 
triangukrum \iiiy\om. 

SumaturjttrumtrianguUm fex quarta f4rs iffius fei* dntr 
qucfit f e i qu^druflum tri4nguli r e x , ^o vero tri4nguU h f ^ 
i 9 1 g m qu4drufUfint qu4tuorfimultri4ngulorum n h o.> p i c\tm s* 
rl fjtmu^c^ 4ntecefi€nti4 aquaUa ;.€ti4m confcquentU aquu- 
UAerunt i ^ritq; tri4nguUm f e x , aqu^U quutmrfradiSlis tri^ 
anguUs nhoypiq9t\Syimu:)^fyft4r€4XQi trifUm erit 
£ \ xorum^ 



1 % JMm etnmdem qndiHor tri^MgitUriim . Cmnq.fit vt ihmm ^iWfmp 

ti . it4 $mtU4Md9m»U : Erit tttmmjhmd tri4ttgitlMm x c g mpbtm 

viiuterf4fgitr4re£Ulime4iHtrifgiir4mmixt4mii^crifi4 . Sed 

€it^demfigitrfinfcrift4tri4ttgnutm ic^mitttuerMt qitimtri» 
flmmimeeefeigititrefytiri4iigidim feg mimmsftqmhit if» 
fmmxc%t9tmmvideUeetfm4f4rte. ^ed^B imft^iU , 
Pttt4mmsdeimdetri4mgmlmmic%efeffmftjm4mtriftmmr4li^ 

qmtfgur4mixt4de9ift4f4r4b9U^-Eft9 &ftt ekteffms ffm^Ut 

ff4ti»V. 
*c Meifi4tmrtri4ugiiUtm fci qmmrt^fmrttttims feg.* & w- 
,rmmfmm4tmriri4tigmlmm fcx qm4rt4f4rstri4tigidi fe k &b»e 
f^tfemtferdomee vemi4tm^ 4d ^Ufmed trimmgmOim yfmtm f c ar ♦ 
qmpd mirimsft ff4ti9 K . Erit^. trimngmlmm x c g mdhme mmisi* 
quiimtriflmm reliqm^fgmr^ mixtm zbc^ii- Sed^mdemfemi-' 
tms r4ti«me , 4tijme ordimevtfmfrm^ eftemdemms trimmfftlmm xcg 
effe triflmmxmimfd4mfgmrf imtrifgmr^m mixtmm a b c g £ df^ 
fcriftf. tteeeffe ergo erit y vt fgmrm mixtm a.bcgf. mimtrft 
qm^4m4ti^m4figmr4fibiimferift4 j tttmmfmmfitrte. ^edeft^A 
fkrdmm « 
' Sierghf4rjAeUtrest4ngeMtsh4bmerity Mfeftrnmeft^ efit 
tri4ngiattmfmh t4ngeittibms ttntetrtmmjreliqmm fgmraet demftd 
fjBr^hoUmtimm. ^moder4tfrof0fitmm&*\ 

Prcfofitio V L 

Arabclla fcfcjuitcrtia cft tfianguli candcm ipfit)afiin,3:eaO 
dcm altitudincmbsbcQtis., 



P 



£|bo pjirab.olfi A B C , cuiuc . 
<Oapneter BD > duie tangcntes 
AH,-CE adi)afim,&tenia F 
B G pcr vetticqm . Dico -paiia* 
bolam ferquitertiam cfle infcr^ 
ti-fibitrian^uli ABCi 

Triangitium enim FBCftct 




du6 



Pfoblema Priimim . 41 

dbotirfliiittiiHm AFB, BGC pcrpraecedefislenmat eil 
vtj.advmim* Erg6irapeziiun. AFGC (cuin tr^wn fic 
truMigttli F£G>addiioeademnriliiieafiiixtacritYt9.advnu. 
£t ad parabcdbim erit /per conuerfionem ratipnis J^ vt 9. ad S^ 
ad.atanguhim ABC» erkvt^.ad^ QuaJiumcrgopardum 
parabolaeftoi3o«taUiimxriangukim ABC efttf; Quaiepa* 
rabolaad inioiptum^itrianguium cft: vt 8» ad tf» ncropc iet^ 
quicenJa» Quoderat&a 

Siparabolatrtsfangentes habaerit ^dcKisad bafim> teniani 
ver6perverticem>&exvniuerfa figura dempiafitparabola» 
dimidiumq;triaflguMfUbtaiigentibusconte«i. Rclkiaa figu- 
fa aNjuq^onderabit ex cuod;»! pnfido » aood ica mrgram tan« 
geMcm iatcndem diatdit ^ vtpars<]iiaradcofiCadimicimaeter^ 
mioaturfitadxc^uamvt 9«ad vnam.. 

bc» cuimsdidme^ 
ter hA CMcipfd* 
tMTdd hm^^mim 
fifftmdiculioris ; 
SintqmtdMMtM^ 
usidtdfim ae» 
cd, ^fttticdis^e^ 
fi^Uj^0fS€hL 
SwH^dtisuie Ute^ 
foli cd'M hiitd 
w cb 4df hd Jlt 
Wj^.ddtmmH}l>i 

cofgitr4m iuiu/m$di{dempt}f4rdMjt,^^emifriMguk vtt^ 
tiMU ebdj^ dqu^tuderdrttxpunif^ h. 
• &utMur di quimjutfdrtiumedrum^qtfdfum^^ 
quarum dh eJfj. Iritq; d h i/^ h i vtj.ttdz. Cum $mem 
wijh ud btrkbmemferftmUcuidriSy f^tttiHtts ^xtf a b c, 

F bcf, 




jps De Dimenfione Pai^abolae 

b c f , ^ftfiff erunt fecundum centrnfmgrduitMis dd funtim9b 
h^ Jime ddfunBumdi. Triangnlumwr^ bdf ^ttdndem cdm^ 
fjnnj^jreodemmodofendebitccnsrutiier exfunS^^ i« (qUMtds^ 
quidem i\ dufUeHiffius i d ; &iff4 db 4d i^mzansem /er^ 
fendicutaris.) I^mffiiHdmAgnitudinesnendquifonderdntex 
h fun£toUbru d i , aliera iffarstmfrafonderj^it • Efio ; ^fTst^ 
fondereht frimo duafortiones mixtf a b e , b c f « Sitqi excef-' 
fusquofrafonderdntyXqudlisffatioK. 

InfcribAtur intra mixtas fortiones figurd ex tangentibus , v/ 
iamfdfefaffum efi . Donec excefiucfortionumfufrafiguram re^ 
SiUnejm infcriftam minor sitffatio JLTtmc enimfigitra infifri^ 
ftd adlfuc frafonderabit triangulo b d f . 

Accifiaturtriangulum d f g quartafarstotitts trianguU d£ 
9x tam. b; ^itq\ tridngub$m dfg aqudU trianguU nf o (cum dmtu 
^* fintfubquadruflaeiufdtm trianguli d i b)&frofterea tridsfgM^ 
lum dfg dd duo trianguU ieai» n£o ycritvtvnum adi. er^ 
go b%iaddmotriahgula lem, nfo y eritvt 3 .ad 2. nemfere^ 
cifroce vtdbadbi. Triangulum igitur b g f /^ duo triam 
gula 1 e m » n f b , exfunSio h aquifonderant itttiicem^ . 
Snmatur iterum d f p quarta fars totius ,d f g , eritqi d fp 
• ^ aquaUduobusfimultriangulisquafunt infra funifd n> c^o. 
^, ( Sunt enim qudrtafortes aqualium triangulorjm d f g , n f o , 

Proftereatriangulum d£p^ddquatuorfimultrianguU\au no» 
erit vt vnumdd2 . Sedtriangulum p f g , adeadem qudtuortri 
dnguU efit vt s . ad2 . nemfe recifroce vt dbyOd h i • Aeqtn* 
fonderat igitttr trianguUm pfg, cumquatuordiCfistriasogtdis 
1, m, n, o, exfuttHo h . Sluafnobrem vniuerfafig^rd itttraftr. 
tiones infcrifta aquifonderabit cum triangulo b p f exfunSfo 
h . Sedeademfrffonderabat trianguU b d f. Necefie igitttr eS 
vttriangubtm bdfminusfitquamtrianguUm bpf^totumfua 
fdrte: ^updefienonfoteH^ 

Fonamusdeindefrffottderdte trianguUm b d fduabusfimul 
. fmionitus mixtis a b e , b c f i t^fonatur excejfus qnofrafon-^ 
derat ^ fqualis/fatiolS. . 

Jccifidtur trtdngulum d f g qndrta fors ifftns d f b .&itt^ 

Tttm 



' Problema Prifnum . 45 

rttmfmmdi» d f p qinBrt4fars iffius d f g , (jrjic/empery dtnu 
vtMtAtttr 4d diquod triMngHli jfuu d fp miniisffdtio K.Tnnc 
enim reUqnum tridmgulu ddhnc frffcnder^Ait fmitn^us mix' 
//> a b e , b c f . Sed oftendemus eodemfenitus 4rgument»y 4ta; 
9rdiftevtfufrdyidemtri4n%ulum pfb fquifonderure ulicuifi- 
gttrfintrafortionts abe, bcf, defcriftf. Necefe ergo erii 
gfModiffa dude fortiones urixtf minores sintqu/hn diiqudfibi m- 
fcriftdfigurd itotumfudfdrte i quod eft dtfurdum . ConSdt a^ 
tji qofodfrofofittfmfiuerdt .. 

Trofopth V !!• 

PAraboIa fefquitcitia cft tiianguli cafldcm ipfi bafiin, 
dem altitudinemhabcndStf^^ 

EfioiUttnprasccdenti Icmmatcpara^ 
bola A B C. cum duabus tangentibus; 
kiteralibus^fiuc ad bafinrv AE, CD; at* 
que EBF per uerticcm ..ConcipiatUFq;: 
diameicr ad horiaontcm pcipendicu-- 
krisi & ablata parabola , dctra^toque 
dimjdio ucrticalis trianguli; accipiatur 
D l teniapars totius DFr& fit DH fcf^ 
quialtera ipfius HI . Aequiponderant 
ergo ( per lcmma prasccdens ) ex pun- 
^o H librse DI, du^ magnitudines. 
Nempe hinc jduae portiones A B C F<E appcnfSe ad pun^um D 5^ 
inde uero tciangulum D B-F appenfum ad pundhim I . Qu;im- 
ebrem DBFadABCFE rcwt ut reciproce DH ad Hlrncm- 
pe Ut j^ ad 1. Sumptifqfanteccdentium duplis , erit totum uerti-^ 
caletriangulum £ D F ad relJquam figuram mixtam iriplufny. 
Propterea (^ut in Propofitionefeitta^demonfiratum eft)parabo- 
kinfcriptifibiniangulifcfquiteniacrit. >Qnod crat propofi- 
ttuiKkfflaiiftfare.^ 




• . • ■ - • • 






Lm- 




44 De Dimciifioiie Parabpte 

: SiduonimcononiraktcrattUDguUpc^ 
in partes «qiiales numero , & roagnitudinc ^duaiiqiie per pua- 
ftafeaionum plaois baC paraJleUs» fiiper ic<aionum arculij 
inteUigaxmircylindriasqucalainffaconosdcia firitutprV 
jsm conus ad fccundum^ ita omnc« cylindri primi conii ad oxor 
joef cylindros iecundi coni » 

Sint duorum €onorum tridngU 
Ufcraxem abc, ^ti^& duo 
€orum Uurd^futs a b> d t^Jtctm 
tur in fartts numero dqualtsi 
men^c im utidem fdOtt€s diui^ 
daturtam a b , quam d e ^^fnUfi 
fdrtes lateris a b fqu^les inter 

fe y (jrfortes d e item aquales interfe . Dit&is ieinde fiffit^ 
guU feSfionum funifa fUnis gh, ii&c. hafi xcfaralUlui 
itemfUn$e m n , o p drciaji d ifaataUeUs ; Ctme^iamtmr ey^ 
lindri ^h , g 1 &c. eiufdemaUuudinis inttjt comnm a b c dejeri 
fti ; itemq; in akero cono. alif eylindri Mfuedti inteUigamtuet 
Bico ejfe^teonus a b c adconmm d c i^ita immesejlindyes ee^ 
ni abc ad omneseylindnseosn d cf > 

Concifiantmr duo coni g a ll , md n^ quenamwrtieeefiof « 
<^d, bafesver^ocirculi^h^mxu 

Jjm ;Cylindrus ah Jidconnm gah, tBvteylindhts d nad 
fOnummdn.(hemfeinrationetrifU)eomnsvero gah adeo^ 
num g bh in eademkaji^ eftvt ag ^^i^ g b ;Jint(fnfterditm^ 
fonem in ^onBrttHione adhikitam)^ dm^dmc^ hoe eB vo 
eonus mdn adconum mcn* €onms deniftte gbh ad eontm 
fmilem a b Q^eH vt cuhus g b ^tdcuinm b a ;Jn} (frofter eomm 
Jfru£fionem)vteuhus mcad esAnm cd, mefevi eotoets mca 
adconumfmiUm A^{^ Sinareexaqt^cylMdrns ^bAd^omi 
zbCycrituteyUtodrus dnjtd^omnmdcl^ £tfermmtamdo ey^ 
lindrus ah adcytindrum dn eritvtconusabc^eomoum^lei. 

Flterius . Cylittdrns etiam g 1 adcylit$dmm mp« eodem fe* 

nitns 



Problema Plriinum# 45 

me^vtcomms abc ^diimmm dt{i&h§cm$0di/emifi9^ ffifii- M 
re^w^MMs^cylimdpmi ah ddvmmm^ dn^ii4^militetmmifcedm\l 
HmmBmdejimemMeiCif^efmimismm^^ erg^ vivmms4dvmmmhmem$^ ri! 
pe x// comms a b c ddammm d e f , iiJi immesjimul eylim^ iimi 
a b c • ^emmesfimmlcjlim^is ami d e f • JSlMfdiJ^c^ 

Lem$m$d XV. 

Datotriliiieomixix>»fubliticdparabolica, em^ 
& alia reifiidiametra parallela compraebeiifo i pdfibile eftia 

dato trilineo £guram infcribere cooftantem ex parallelogram^ 
mi$aequealtis,Qii«fi^iradeficiat atrilineo mixtominoci dif^ 

fcrentia quammqiuscumqftdatamagnimdo* 

EjiolinedfdrdbiUcd a 
hc ^cmiustdngens cdjCf* 
didmettomqt^dijldmi^i a 
d. Z>/V# M/ri nilimemmtim^ 
imm a b c d • defirikififfe 

fgmrdmcomfidmtom ex fd^ 
YdUeligrdmmis fqmedltis^ 
qude fgftrd deficidt a triU^ 
meomixti^ mfimori defefht 
qudmftffdtimm quodctm^ 
qmeddtumK.. 

Secetur enim d c Hfdri^ 
dmimx: iteritmtq;fdrtes H 
fdriimdiuiddnturin h & 
» p femferfi boifdt^^ 
mecvenidtur ddfeSiomem 
dliqmdmfmtd de ^eimfmi* 
diy vtfdrdUeligrdm. a d c, 

mimusftffdtiolL (^iddmiemi^cferifoJii.fuiei.Sienim 
cif^lcdimrfdrdllelogrdmmfum a d c , iMiffffercomtimudmti^ 
fi&a^mmfemfadeirdhiiMidmidimmi ergUdmdem remmme^ 

bii 




Pl. 



4ff De *Dimenfione Parabolig 

i^// ae mmtis quoUbet datoffdtio .) \DMcanturdeinde exfmn. 
^isfiGionum refff e f , h g drc. stquidiBdntes iffi d a ^perfnn 
V 0ddutem iy b^r. *vbi fardlltlffecdnt fdrdboldm^ dncnntm 
1 g , m n (^r . dquidifidntes tdngenti c d . EtfdSum erie qttdi 
wfortehdi. 
^e. Fri* PdTdUeJogrdmmum enim c o , equdle eft iffi o p» cf nddito 
communi or, eruntdtto co, OYydijudlid iffi vc^^fiue iffiii: 
ddditoqi c omunii l y er un^ trid co y or, Tlyfqudlidiffii^hoc 
eftiffiiXy ddditoqi comuni tz . (^ficfemferfrocedendo , ^rmM 
dtniq. omnidfimulfdrdUelogrMmd cortzybia dqudUd iffifn 
rdllelogrdmo a e . nemfe minordffdtio L ilinr/rp if i/Ji^ miner erid 
dtfiStusfgurdinfcriftf ex fdrdllelogrdmmis dquedtts comfo* 
fitf^ Atrilineo mixto abi;dj qUdm fitfrofofitnmffdtinm K^ 
£moderdt(^c. 

CbroIIarium* 
Hincnotdbimusy quod eodem frorfus modo ^ ettdettufioferd- 
tioneyfigurd etidm circumfcribitur ddto triiineo mixtOycenff dns 
ex fdrdlleUgrdmis equedltisy itd vt excejfusfigurd circnm/irif 
tdfufrd iffum trilineum » minorfit quocunqueffdtio ddto tL 

. Lemmd XVI. 

Si parabola tangentem habuerit : & infuper duas redas dia^ 
metro paralielas , opse duo triiinea abfcindant fub tangente, & 
linea parabolica comprar henfa s Erit figura ex paralleiogram* 
itiis aequealtis confians in maioritrilineo defcripta, ad figuram 
eiufdem fpeciei in minori trilineo defcriptam, vtcubus maio* 
ris tangcntis ad ci^bum minoris • 

£Sfofdrdbold a b c , cuius tdngens c d ; e^ didmetro fdtdUe^ 
Idfii^trdqncA a j ef; vtfdnt duo trilined mixtd abcd mn^ 
iusy (^ (h c c minus . Dicoyfiinntroquetrilineoinfcribdittrf^ 
gurd eonfidns ex fdrdllelogrdmmis dqudlibus utrimque ntttttero^ 
( ut ipfrdcedenti Lemmdte exfofitum efijfigurdm triUnei a i> c 
^^^dfgKrdmtrilinei fbce, effeutcubtu dc ddcubum ce. ' 

. . Cencifidmuty{ddeuitdtMUnnlinedrumrnultitntiinem^con 

fufio- 



FtoblemaPrimuffl^ 



47 




fitfieitm} tri- 

tmiMlitm gcc 
emmfMftrtia 

tereeft4(\>Ct 
trdu^erriytttf 
Ji idem qm»d 
fofitmm efifiA 
Jtgmj\i'\\. tri 
UneMm^. fhc 
e e^ idem rm 
trtti»eemxi\\, 

Infcrihatmr , 
immimMtro^Me 

trilinto abcd, ^ mnli, {^mtdquidemrefrdfeHtdt iffmm 
fbcc trifldtMm)figmrMConft4msemfier4ileU^4mmis 4^Mt4U 
tis i <^Jit idem mmmerMs fsrdUk^^mmmtm in Mtro^. triiime», 
ItttelltgMtMr etiMmcoaut tCMtMs Mertex c, _fiMt\ > <^ di^meter 
hafisfit ^hinc qmidem ad, imdemtrehi. Sintqmtimfimgmtis 
eenifegmentiseyUndri(cimt4l0i op, qr^f. 

J4mf4r4Ueie^4mmmm bp 4d (diy efiMtrtifm br 4d fp, '''""^ 
bec cft Mt qM4dr4tMm rc 4d cp; hoc tB Mt]qH4dr4tMm rc V**^»^ 
fM4dr4fMm pu; hoe eft mt cjUmdrms qr Jui eyUndrmm op.*^""* 
Eodemmedo ycritfjadiUeUgrMmttMm xr md (djMt cylimdrMs 
yr a/ ud. ErgoerMmt dMofimoMl.f4ridltUgr4mimu bp, xrj^*"^^ 
a/ idiMtdMofimMlcyUmdri cp, yTy^dcyUnc^mmnd. Pro~ 
ttdemdo it^tfntfemftr hoe mudo, (jr dtmi^mc eormfonemdo,erit tt 
t4 infcriftafigitr4 exf^aUelogr^mmis confiams im triUnto a b 
cd, 4df4r4Uelegr4mmMm fd, MtemimesfimomleyUndri,qmiim 
fone acd, 4dcylindrMm ud* 

^mflims ifaraUelegramtmmm fd ^ ni tomeforit^mh^tt 
fatiotttmttxrationereiif {^ 4dQZ,siMeqM4dr4ti pc a/z1 
(/ir/r/ MMs dmafigmra ,fede'm4 e^mdemf^r^olim tr^msUtam) 
4imtqm4dr4ti pu .«/ zK; Etexr^tiomereSe dp 4diz.Eff 
ergof^r^Uehgrmmtm fd <«i/ nl «/ eyUmdrms u d W ki. 



4t DelXffiettfiooeFdraboIx 

DemqnepardUehgrdmmtm n i ddtotMmfigurdm iafmft^^ 
^tnif tratrilinenm mnli, ejiiit eyUtubrus Vi sd »mms cyHmdnf 




^. *•* infcriftos intra conum h 1 i , Proftered ex fqmo *f 

,*^«r« fdrdlUlogr^mtmisconlfjmswfcriptdinmdioritrmmep a bcd , 

>%«r« . Mdfignram ex fMrallekgrammis infcriftdm im miemi triBmf 

mnlitiitomnescjlindriincono acd ad omnes c^m^s im r*- 
*«».»♦. »* h\i. Nempentconns acd adconnm hli» hocefddnmtm 

g c c (qididem eji . ) iyTflw/* «/ f «4w d c adeubmm c c . .^«•/ 

cratc^c. 

Lenma XVtt* 
SiparaboU tangentem habuerit, &infuper<laa$ diamctio 
parallclasre<aaslincas,qu5 duo trilinca mijttt abfcindaimE- 
' runtinterfeabfciffatrainca vtcubifuarum tangeatium. 

FJhfdr^olazhCtndwtiam^ 
gons cA : (^ diamem fdn^U 
fk ntra^ d a , eb . £>icotriline. 
mmmixtnm tibcd adtriiimenm 
mixtnm b c e , fff»f ^f^t tan- 
gentis d c, dd cnhmm tagemtis 

■ ce. 

. Siemmirinmeftysitahernm 

iUtmm^sifo^ilt ejty mains qnM 
nthalfeatdiifamfroportionemad 
reHqnnm ,• (^fonamus iHnd efiji 
zhc6,mainsqMamfn$defiede' Cj 

heretexciffii Fl. 

Infcribatveintr^pr^^emm ab c d yJ^*^^ cxf^dUehgr* 

mis fqnealtis conHdns i ita ut atritinio deficiat mintri defeau 

Um. 1$ quamsitffatium\ (hfeAUtemfierifopoftendi»ius)ffa$eht- 

queadhucfigurainfcrifta jid reiiiptum fritiueuiH bec mattri 

tationtm quamcubus d c adcuhum ct. « /• 

InfcribaturintridUerumtriUneum hccfiguri eiufiemffi 
«rt ,& eiufdem numerifdrdkU^dmmorum cumdefcriftd tn~ 

trd 




: Preoblma Friiiitiiti# 49 

tri£n€$im abcd. EritngofgtfrdMfcrifumlmea^hcd 
^ figurdm inftrifiAni triUncO bce .vrculMs dc 4dc$ti$m ^^f^^ 
c c « Sed 4dd€mfigmra imfcir^d triUmi a bc d dd iriliHMm '^^"^^ 
bce b^t mdiirem rd$i9$um q$tdm cmiMs d c dd ce. Ar^ 
mms ergo eft friii$$4$$m bce ^Mdm imfcrifed fti figmd ^ teimm 
/mMfdrtt. Snpd eft imfoSihile ^ CeniiMtergefrefeftuim^ 



Profofim VIII. 



p 



^riJ>o]a fe£;)uireitia eft xriang 
eaodem,aImudintm babenns « 




JEftoparaboh AfiC> cuius 
diameter B £9 tangeines vtro 
AF) CF) produdo? eoufque» 
4I01WC occurram jpfis A D^ C f^ 
diaftietrbparallelis. lutigannjrq; 
Ted^linefAB, BC.(lk«infi- 
gura omiiori^fint.^ DicoparabolS 
Q:iai^uli ABC efse feiquilrrtia. 

Eritcnim ABCDadtriline» 
um BCF^ nt cubus DC ^d 
cubuai CFt nettipeutotAoflil ^ . 

imum./cumenim fitut AE ad * **• 

£C,ita DF ad FC, erit DF aequalisipfi FC; ciiMi% 
DC oaupluscubi CF.;^ Itemttilineum CBAH ^txy- 
linetim BAP,eftutoAoadiAium. Coniundini erg5 erum 
duotrilinea ABCD» CBAH, ad ^tiudi ABCF. ut 
odo ad unum • £t diuidendo bis^erunt duo triangula A F D » 
CFH^ ad^atium ABCF, iK^*atlunum. Qu^mobrem 
iriangulum AFD, fiue AFG adfpadum ABCF critut 
}9 ad mHUA ; & ad parabokm er ir uc ^ . ad 2. oei ut & ad 4« Pro* 
ptereaparjkboUieritailiriafigulttm ABC ut4.ad3. Nem^ 
peiefquitertia^ C^d^atpropoiitumdemonftrarCi&a 



\ 



y> De^^Dimehfione Parabolx 

. . • 
Ltmmd XV III. 
S\ fuerit v« prinu magnitudo ad fccundam,ita tcrtta ad qy»- 
tam ; Et hoc quotiefcunq; libucrit « Fuedntqi omncs primx in- 
ter fe , item omnes terti« magnitudines inier fe asquales • Eruot 
omnes primae fimui ad omnes fccundas« vt funt omnes tcrcios (i* 
mul^ad omnes quartas magnitudines « 

Efio vt a fnrimd dd b ftcunddmMd T T T F 

C ttftiddd d ^jMdrtdm. Etiterttmvi aI £1 li i 




o"p 



e frimddd i fecunddm^ itd g tertid „ ^- j^« - 

dd h ^MdrtdmiEt/ic quotiefcunqy Itbue^ | | * i 

rit. Sintqyomnes frimd a, e, i, &c. 

ittmomnestmid c, g, mj&c.inter n n fl n 

fedqudles. C U<iLJjvJJ Li 

Diceomnesfrimdsfimulddomnesfe^ D nHnKn r% 

€unddsJimul,itdeJftvtfu9^omnesfimttl 1] TI Q 

tertidy ddomnes qudrtdsmdgnitudines . 
Sluonidmenim conuertendo efivth 

dd z itd d dd c. Itemvt f dd c; 

fiue dddqudlem a» itd h dd g^ fiue 
^^ w-^^ c; eruntfimul bfdda^ vtfunt d h fimuldd c. Hoc 
u. modofroceaendo , oJlendemus omnesfecunddsfimulefile dd a , 

vtfuntomnesqudrtdfimulddiffdm c. Iffdvero a ddomnes 
t^qmm-^frimdsefivt c ddomnestertids(funtentmdquefuhmultiflices) 
tl . £fg0 ex fquo omnesfecundd dd omnes frimds , fnnt vt omnes 

qudrtdfimulddomnes tertids • Conuertendo igi tur coHdt quod 

erdt frofofitMm demonfirdre . 

u 

m 

Lemmd XIX. 
Si parabola tangentem habuerit ad bafim > ex alia vero par- 
te redam diametro parallelam • Erittriangulum fub tangcntCi 
& parallela diame&ro » ipfaq; bafi compraehenfum f ipfius para« 
bola^triplum* 

EJI» 



Pfoblema Primum 



51 



2) T H L 



Q.t 




A a 1 M. 



smMstdngen^c d, f^dl^ 
UUdidmemfit ad> i>^ 
€$tri4ngMlum zdkC effc 
f^iA$U $ffius zhc ^ tri 
fUm. 

Sietnmmm efitrifUm 
f^db$U jfercomicrfiane 
¥4tidMs^noH eritfe/qtiidt . 
terttmtrilinei z\ycAs& 
froftered fdttfUcdtih dn^ 
tecedente)totHmfdfdUe- 
Idgrdsnmttm ae non erit 
trifUm trtUnei abcd* 

TriUnenm ergo a b c d ertt velfUts , vel mintts fndm ter/id 
fdrsf^dUelogrdmmi a t . Pondturfrimum efiefUsqHdmtertid 
fdrsy&fitexctffiiifqudleffdtimn K. 

InfcribdSuritarktrilineum a h cd ^figurd confidns exfdrA 
Utogrdmmis dquedtisy deficienpfue dh iffo trilineo minori dtfem 
ifu qudmfit iffumffdtium K • Etinfcriftd idmfit eiufntodifi^ 
gurd . Erit ergo ddhucfigurdinfcrijftdfUsqudmtertidfdrsfdi^ 
rdlUlogrdmmi ac* , 

ConctfidturcircdreSfdm zdy ctrcuUs^ quifithdfiscuiuf 
ddfk eem virticebdhentis in funClo c . &fufer eddibdfi inteU. 
ligdtur cjlittdrus a e eiufdem dUitudinis cumiffo cono ;fecitf^ 
quefit tdm conus qudm cyUndrus fldnis bdfifdrdlleUs ferfingtt 
IdsreStds fg, hi> Im c^c.duifis. Concifidntur etidm intri,^ 
conum acd cyGndri dquedUi po, oi &c. t.fexti. 

Idmfic z Pdrdttelogrdmmum a f ^^ n d , efivtreHd da dd "^^^^' 
o n ; nemfe vt quddrdtum d c ddquddtddum c o ; fiue , vt qud- 
drdium d a ddquddrdtum o g , ncn.fe , vt cyUndrus a f ddcy^ o*» fimdr 
Undru^ po ^Etficjfemfer. Suntqs omnes ffimd mdgnitudi- ^^^ 
nesfo^dUsfdrdUeUigrdmmo af, ^ideoidtftudisinjterfeyom* 
nesdutem terti^ mdgnitudincs ^equdlgs cyiindro . a f , dtque ideo-x 
interft\ Eruni ergo omnes frimdftmul^jtfic efifdrdlUlogrdm^^ 
• * * G 2" mum 



^z De Dimdnfione Pafabolx 

mttm a q » ddomntsfecunddsJimMltn^mftddfgMrMm imfkrifti 

in trilineo a b c d , v$f$mt imnfs tertikjhmnl, rumpe cjiUmJrm 

^C[^4d onmes quMrtdsfimttt^htK eji ddmmtiscyBndnsitoSrjic^ 

num a c d dcf^triptos . Conuertendo igitur '; erit figurM trdineo 

infcriffdadfOTtdlelogrammum^iQ^ vpmnes- cytindri it$trMC0- 

num ac d ddcylindrum a q . PdrdUelogrMnmum vero zq dd 

fdr^llelo^d^mum ae efivt ^o^dd ^^^hoc ffivt cyUi$drms tl^ 

ddtylindrnm a e . Proftered ex dqu$ yjigttra infniftd t» trili^ 

treo dd totum fdrdUelogrdmmum a e, erit vt 4mnes cylintlri im 

copoinfcriftiddcylindrum ae. SedjfgurditrfcriftaimtrUittgt 

e/fexidmdiitis)flufijudmtertidfdrsfdrdlUUgrdmtni zt,tt 

ffi omnes cylindri in cono defcrifti eruntfltrfqui tertia fart ey. 

lirtdri a e , nemfe maiores quim conus a c d . fort vidtlicttfm 

toto. ^od eJHmfoj^Bile . 

Std fotuttmtt tttmc tfilinettm^ a bcd eftminus tfuam tettid 
fartfaratttkgrammizc;ftq;deftiius tqtfydit/fal^K^ Cir^ 
^^'^ cumfcribaturtrilineo zh c d JignracanJfans tttfaralUlagt^ 
* misaqnealtis exctdtnfq;mino ' ^- ^ ^ ^ ' " 
tritfigttra circnmfcrifta adifmc 
Itgratomtoi zc . 



V m 



Cotttifiatttriftrrnmcir 

careifam ad circuUts 
ffO iajlconi , quivertitt^ 
Itaktat c; ittmq;frt ia* 
Jicylittdri zccd eiufdS 
altitudinis cum ifjoct^ 
totzcd. 

Jntelligatur infufer 
circa conmm defcriftaf- 
gmrafoUda comjianstx 
cylindris aquedltis aq, 

Idm farailelogr. 




mum a f da farallelo^awmum a q (ob dqudlitdtem) eBvtcy^ 
lindrut zdi^ ddcylimdrum zdc[X . AmfUmt. PdraUtUgram 



mum 



Problema Primum^ 55 

mMm ^h dd fMMUtl0j^4mmMm li ejlvtg f^JfMf zdmt iq; c/##fS. 
memfe vtMddfdtmn d c ddqHddeMmn c q » /W^ vt qetddrdttm^ f^'*'" 
da, t/r/ t^^ddftiddrdtmn gqi ttetnfevt^ylit^us gh ^rf^. ^ Tj^ 
Ihtdrum g i« ^^^hecmedefemfer . Sttmqi ^mnesfim^ild^ tmdani^ 
timfrimdfndgmtMditiesdqtidlesfdr^tegrdtmmzft &idA t^ , ^ 
interfe: item omnes tertic dqudles cylindro ^if^&^t idinterfe; 
ergo erunt omues frimd pmul^ hec eB fdrdUelogrdmmttm zt^dd 
omnesfecunddsjimuly hoc ejlddfigurdm triUneo circun^cfiftd^ 
vt omnes tertiafmul , nemfe cylindrus zt^dd omnei qudttds 
fmulynemfeddcyUndroscouum acd circumferibentes . Coth^ 
uertendoigitur ytrit fgurd circmufcriftdtriUneOydd fdrdUeh-* 
grdmmum ae^vt omnescyUndrictrcttmfcribetttes conu ddcy^ 
tindrum a c • Sedfgurd triUneo circumfcriftd rmtteir efl qtthn 
tertidfdrsfardUogrdmmi a e ; ergo etidm mmes cyUt$dri circu* 
fcribentes conumminores erunt quam tirtid fdrs cyUmbi ae j 
Nemfe minores cono a c d . Totumfudfdrte : qmd ejfe tten^^ 
teft . Tridngulum ergo a d c ifftusfdrdhoU omttinittrifbtm erit. 
^odfrofofttumfuerdt. 



p 



Profofitio IX. 

:ertia eft triangiili eaa4em ipii hiSatk 



eandetn alticudinem babentis • 



Eftoparabola ABCcniusdia^ 
meter £ B , triangulum infcriptum 
fit A B C , Dico parabolam trian- 
guli ABC efTeiefqitftertia]»; 

Ducatur enim tangens CD, & 
fitreaa AD di»mmo asquidiftaittt 

Erit ergd per prasccde» lhiima# 
triangulum A C D porabolar cripUii 
& propterea erit parabola parces 

Suatuor earum, quamm a-iangutum 
k D C eft duodccim^nempe quaUii 




trian- 




^H*-^ 



De Dimenfione Parabolf 

_.unguliim A B C cft tres . ( triangulum enim A U C «quale 
cfttriangttlo EFC,cumvtrumq; duplum fit trianguli EB C, 
«rgo triangulum ABC qprtoparscrittotius ADC J Con- 
ilatcrgoparabolamadinfcriptum fibi triangulum eifc vt^. ad 
j» N^mpefdquitertiam. QupdScc. 

Prpfofifio X» 

AraboIafefquitertiaefttrianguUcandem fibibafim, cao- 
demq; altimdinem habentis • 



P 




Eftoparabola ABC 
cuius diameter B D . Di- 
coparabolam ABC in* 
fcripti fibi trianguli cfTe 
fcfquitcrtiam • 
. Compleaturparallelo^ 
grammum ADBE» & 
niii parabola fefquitertia 
fit trianguli fibi infcripti» 
neque (Tumptis dimidijs^ 
ftmiparabola ABD fef- 
quitertia erit trianouli At 
B D ; neq; eadem femipa- 

rabola ABD erit 2 .;ert. parillclogrammi ED, fedvelplu^ 
vcl miniis qudm 2. tert; ciuldem « 

' Efto primiim fi fieri poteft femiparabola A B D magis qua 
2. tert.parallelogrammi £ D s & ponatur exceffus «qualis fpa- 
tio K* Ipfiqjfemiparabolaf figurainfcribatur conftans expa- 
rallelogrammis «quealris ( more apud Geometras vfitato, pro- 
ut faftum eft Lemmate XV. ^ ita vt difFerentia inter figuram in- 
fcriptam , & ipfam femiparabolam minor fit fpatio K. Tunc 
enim infcripta figura adhuc maior eritquam i^tcr.parallclogra 
miADBE. 

Duca- 



Problcma Primuiii • 55 

Ducaturcircadiametrum A C femicirculus AXC» com^ 
pletoq; redangulo ^ fiue quadrato A F X D • ducantur G L, H 
M, 10 peipendiculares ad AC,& compleantur re<5langula f . 
DL> GM, HO] Tumintelligaturfigura AFXD circumuer 
ti circa axem A D > ita vt quadrans A D X hemifphaari um de-* 
fcribat,quadratumver6 AFXD, cylindrums & re<%anguLt 
in quadrahte infcripta totidem cylindros fitciant in ipfo hemiA 
ph acrio comprajhenfos . 

lamparalielogrammum BGadPD, eftvtBDadGPi 
fiuevtrei^ngulum CD A ad redtangulu CGA>fiuevtqua- 
dratum X D ad quadratu L G > fiue vt cylindrus X G ad L D. 
Et hoc modo fempcr • Suntque omnes prima? magnitudines x^ 
quaic9 paraUclogrammo BG> & omnes tertiae aeauales cylin- 
dro XG. Ergoeruntomnesprimxfimul» hoc eit parailelo- Lem.it. 
grammum T D, ad omnes fecundas fimul > nempe ad figuram 
infcriptamin (emiparaboIa> vt funtomnes terd^ fimul > nempe 
cy iindrus V D ad omnes quartas Gmvl^ hoc eft ad omnes cylin 
dros in hemifphaerio infcriptos • Parallelo grammum vero TD 
ad £D eftvtcylindrus VD ad FD» ergocxaequo» eritpa- 
ralleiogrammum £ D ad (iguram in femiparabola infcriptam 
vtcyiindrus FD adomnescylindrosinlpfo hemifp)iaerio c&« 
praehenfos . Sed parallelogrammum £ D minus eft quam (c(* 
quialterum figura? intra femiparabolaminfcriptae i £rgo cylin^ 
drus F D minor ci it quam fefquiaiter omnium cyiindrorum 
inhemifphxrio d^fcriptorum. Quod eft abfurdum. Scimus. 
enim didum cy lindrum hemiiphaeri; eife fefquialterum « 

• 

. Eftodeinde('(ifieripotefi)femiparabolaminorquam2.ter& 
ipfius parailelogrammi £ D • Ponaturqs defe^aos xq usdifi fpa« 
tio K. 

. Tum ipfi femiparabola? figura quasdam circumfcribatur, con 
ftans ex parallelogrammis a^uealtis (more folito,vt fiidum e(l 
in Leoimatc X V* |eiufque Corollario) ita vt ditferentia inter 
circumfcriptam figuram ipfamq; femiparabolam minor fit fpa* 
tjoK« Tunc enim manifeftum eil i quod figura circumfcripta 

adhuc 




JLemM^ 



5d De Dimetifioiie Pirabolae 

adhuc minor ehc quam a. 
fer. parallelogra tnintED . 

Fiat cireu riiametrum A 
C femickrukiS) vt inde^ 
lcriptione pr^cedentij con 
firu^onis , complecoque 
qutdroto AOFD, perfi- 
ciantur rcliqua recfiaDgula 
FL,GM,HN,IA.cir. 
ca quadrantem defcriptat 
Tum reuoluacur figura AF 
circaaxemAD, kavtfo 
lida genereisur ram di^a : 
nempe hcmiiphan^ium ex 

^uadratite,cylindni6etqimdrato AFt totidemque cy lindri 
4uot rwaangttla enmt ipfiquadranti circumfcripta • 

Utn pdtvUtiograrmmim B L adfe i^Qmi^ft vtcylindntt br 
A\xi ek F L ad fe ipfttm* AmpHus • Paraiielogrammuifi QM 
ad PM;eftviQJwad LP;(aieB D ad LP,fiuevtrcaang. 
CD A ad CL A, fioevtquadrammFD ad LG, (iuevtqua- 
dratum R L ad L G; nert^ vtcylindnis fadhis ex R M ad cy- 
lindrum ex G M : & hoc ntodo iemper « Suntq; omnes priaa 
magnimdinesaqualesparailelogtammo BL, omnefiq; tertije 
^uales cylindf fadto ex F L . Ergo erunt omnes prinue (imol 
rtempe parallelogr ammum A B ad omnes fimul fecundaj^ ne- 
pe ad figuram femipar&bolr circumficriptam , vt funt omnet tef 
tia? fimul, nempecyiindrus ex O D £adus,ad omnes quarcas $ 
nen^adcylindrosliemBrphderfo circumfcriptos • Sed paralle- 
logrammum E D tndgis eft quam feiquialterum figurf circum- 
fcript^ ad femiparabolam , ergocylindrus ex OD magis qQdm 
ftfquialtereritftdofrtrtes cylindroshemisplMerio circumferip- 
tosn Quod eft ablhfdtim: Scimus enimcylindrum hemi^rfia^ 
riOcfrcuttifcriptUiti iprtus llemifplmijeflre fefquitltenml^ 

Patet itaq; paralklogi^ £D fcfquialter u efle ad leimparabo- 
httt ABD; &id€t)itmipaTab.l^^ ABE^ 

oyA- 



55 



QyADRATVRA 

P A R A B O L AE- 

per nouam indiuifibilium Geometriam 
pluribus modis abfoluta • 



ACTENVS de tUmtnfitne fdrMm 

$Kore MtiqnprMmdi^utHfit i Meliqtitim 

tBvtedmdemf^iAtU mettfnrMt tmiA 

tptddtm tfedmhrjAiU TMtUttt d^edi4- 

mifr;9fe fciUcet Geametrid ImUi/ifihi' 

I Uumy f^hecdiiieTfism»dij: Stiffefiti* 

enimfrdcifiiis Themmdtih. dmiqimr^ 

\ tM»EitcUdist qnitm Arcbimedisy Ueef 

dtreins iHterfit diMerfi^misfint , tmrS 

tfiexvMeqiie^ieoriimqiiddrdtitrdmfdrdhaldfMciU negetieeUci 

/A^; & vice verfd .quafiedfit ctmmmie qmedddm vincmUim 

vtritdtis , Peftte enim ftiidcyUiidrits iiiferifti sAi ceni triflms . 

fitt bitufeqmtitrfdrdheUm inferiftifihi tridngnU tfiefipjni' 

ttrtid:Si verh mdnitfrfmittere cjUndrnm ii^eriftdtfibiffhde- 

H • tdeefit 



k6 , De Dimenfione Parabolas 

^d effe /ifijmdhcrmn » €9HtinU9 fMt^eU q$udt4$mM orfet' 
tur . Eddim cendMdiiMr frfpjitsi dcimnftr^uUMC^ quMfr^dt 
€emrum ^Mnitatis €om f$fit$tm eftim dXCyita w fdrs qtu 
adfvertietm tfiy retiqufjit trifU . FUfAk§U mm mUms qumdrd- 
tuf eHdmfuff^mendoJ^funumd UueuffirAiiHfrimAtemUttio 
HedefcriftA/cjrjireiidqUdiHitiumtfireMiUti^miSy (smfrfkem- 
fum yfubtrifUm ejfefrimi circuli . Cemtri ver\ l fttff^JftJt fd- 
ruboU quadrdtura^frfditid omuid TheeremdtdfdCtle demMnfird 
rifofiiHt • ^od dutem hfcIndiui^iUMm G^metrid n$umm 
femtus iHMtntum sity equidrm^mm^di^fi^dff^^ . Crfdidrrim 
fOtiusvitetesGeomeJfdshdt thttidfv/istH Ussemtime TbdO- 
remdtum dipcdlimorum , qudmfUdimdrmomlfrdtioHihus dlidm 
vidm mdgis frobduerimt ^Jiut ddoccultdHdum dttis drcdHumJl^ 
me ne vlU inuidis datrdSioribusfreferretur occdjio contrddicen- 
di^^kquidtft,ctrtmm4BhdHcCroiHitrim mmrmmeffefrom* 
mentione comfgndimH^iHHMmird qmdji imffrfcrmdhiUd^heo^ 
remdidyhreuibusjdireSfisydffirmdtiuifqidemonJhdtioHibusceH 
frmdre; quodfer doQrindH^ dntiquarum feri minimk fotcH. 
Hdc enim eBin Mdthemdtisisffinetisvidveri Regidyqudmfri 
mms omnium dferuity dr ddfublicum bonum comfldnduit mird- 
hiliuminuetuorum mdchiudtor Cdudlerius « 



p 



Trofofitio XL 

t 

Arabola fcfquitertia eft trianguli eondem ipfi ba{ijn»& eaa 
dcmaltitudinemhabentis- 



Eftoparabola A B C cuius tangens C D, & diamctro aequi- 
diftans fit A D . Perficiatur parallclogrammum A E j & circa 
diametmm ADiittelligaturcirculussquifitbafiis coni cuiufda 
verticemhabcntisiti puniflo C^/Scitem fit bafis cylindri alicu-^ 
ius A C £ D eiufdem aicitudinis cumdiiSio cono « 

Ducanruriafnquaelibet rtda FG p.arallela ad AD, Scper- 
jpiai;n intelligatur firan&e plaimmparallelttm circulo A D* 

£ric 




Pitoblfna Ptimum* fj 

Erit erg6 F G ad I B vt rc<aa D A Jid iB 

hoc eft vt quadratiBn D C ad quadra- ** f^, 

cumCI,fiiievtquadratumD A adlGk P F !£ *!''••. 

hocettvtcirculusDAaacirculumlG />s^^ i I iMi.M 

nempevtcirculusFGadeundemlG* / /^ / ^- 

Ethocfemper; fuoique omnes prima? 

magnimdines aequaiesTe^ DA. dc 

ide^ inter fe;omnef etiam tertiae «]ua- 

les circulo D A , & ob id inter fe ; ergo 

per Lenuna 1 8» eiunt omnes prim^ 6r 

mul y nempe paraUelograromum A £ » 

ad omnes fecundas fimulf nempe ad tri 

iifteum A B C D>vt funt pmnes tertiae^fimul^nempc cylindhis A 

E^adomnesquartasfimttlhoceftadconum ACD. Eftigi- 

nirparallelo^ammum A£ triplumtriliiiei ABCD. Sump* 

ioquedimidio»eri(triafigttium ACO fefqpialtehiih trilinet 

ABCD;&perconuerfionemrattodis»erittriangttl^^ AQ[> 

triplum ipfius parabolae • Propterea # ex demOni&atione pro- 

pofitfonis 9. erit paraboU inioipti fibi trianguU fefquitema • 

Quoderat&a 

Jlid qu^que rdtionc pdr4A$Um qmAdfdbhms ^ dtmonfifMis 
ffius y qmafnifotmt treuitdte , indiuiJUfilittmffitteifqs . De* 
ttiitdhimus Aute db immeufo CdstjUridttf Geometrid ocedno ^ mi^ 
otOTidtuldcidrddftttes terrdm. ^i volet^ hdc omttid videre 
foterit (infintedicdm , dm imfddgo fj circ4 ntedsmm fecundi 
tibri Ceometrid Imdiuijihtium Cdmdlerq • 

Lermmd XX. 
QiKuiraca omnkm partium cuiufcunqi re^ lioea? fubtripU 
fimt totidem quadratonim totius . 

EBoqmdtihetre&dtimedzh.DicoommUJlmmlqmddrdtd omo^ 
mitmmfdrtimmre£fd ab eJfefmbtfifUtisidamqmddrdtormmtittf 
demre^dtimed Ah • 

ff j Fidt 



5< 




De I^meniioiie Parabdla^ 

TiMtmmqMddrdtmm acdh^iUf^J^f;dut^ 
mMr^ ad. ^pmmmMw ffftrd circdtMxt^}^ 
' dcmec im emm Ucmm redeMtnmdtcefit mr&meri • 
' Itmmifeftmm e B y^meda quddrmio cylimdrms c 
h dtfitHtetMr^jm^^utoveib abd cemms d 
ah, qmiveruccmhdbeiitim a. Dmcdcmrimm 
^mdlwet e fpmrjdleU iffi c a > eritq; a f , fim 
f g (ftuuemm dqmaUs) vnmcx imjinisis fmru 
bttsteumc ab. 

/Mi i qmmdrdtmm tefius ab , mdqmddrdtmm 

fstrtis ztj efi^et dqttdUtdtem y vt qmddrmsttm 

«•DiM- tldd fgymemfe vt circulms didmetre t\fd^ 

**'"^' Itusyddcircmlmmdidmetro g i • Etfic erit ftmmer . Sttmifi fru 

md mdgtHtmdines fimgmlk dqttdlerquddrdSb ab y^ tersid fctt^ 

^'^^* fer dqudtes circmh dh« £r^^ emttes frimf fimmt ^ hoc efi t§i 

quddrdtdRned^h^^quetiffdh^etfdrteSy dd omtttid qttddrdid 

fdrtiumy erumt vt omnes tertiffimttt^ hoc efi vt^ylinduts chdd 

omnesqudrtdsfimutynemfeddcommm dah. Smntirgoioiqmd 

drdtd dticuis Uncf qmetiffd hdbetfdrtesy dd omnimqmdskdSd 

fdrtium iffius vt cytitulrms c hddcomtmm d a h , nen^e sriftdSo 

ciuersendo jconfidtjfofofittmn quod dmmnSrdttdtmfitertu d^^ 

Lemmd JX X I*. 

\ Omnia re<5hingula y quae condnemur ftib altqua reda linet 
cumfinguli&fuispartibusy&reliquis partibus, iubfeiqaialfen 
funt totidem quadratorum du£dem redae liner * 



Afiumftdpdcedentis Lemmdsis figurdy dcceftmmfit im reSd 
jlb quodUbetfutiffttm f . Heitdngmmmfub b a f tdstqmhn vnd 
teffdtined , o^fitb fb. contentumyeritvsmmexomstibstsfrd- 
diiiisre£tdngtiUs(vnmm enim tdsus comfotutttr ex totd a b, ctnm 
fttrjte a f ; dUejtum vero efi fb , mimirttm retiqudfdrs . ) 

Xeffdngtdum dmtemfrddHiumfub b a itdntqmdm vttite&d , 
t^fub f b eontentum ^ idem eft , ol dqudUtdtemidtertmt , dc is^ 
Sdngutmm e i 1 . EtbicftmfJtr virmm nrisMc modo , vbicmttqi 

'fitf. 



Problema Primiim . 59 

JttfnHBum f . Sed omnid YeaamguUfiArt&is intetceftis im 
irafe^a c a h d (efndUnm vnd ejt t\) &fnt reli^nis j^jnMinm 
vnn ejfils vnd emm omnitns qnddrdtis intermedi4»nmfe£Henn 
(qndinm^nd£ft fij nqndntnr^freffer v fecnndi elementern) 
dmnikns qnddrdtis dim$didrnm > qndlinm vnd eft fl . Omnid ve 
r^qnddrdtd imtermedidrnmfiitionnm (qnnUdm vnd eft fi} dd 
omMdqsiddtidtd dinrididrnm ( qndtinmn/nd eS f 1 > )fimt^t vnt^ pf^4, 
dd ^. SUrgo demdntnr omnid qnddrdtdintemudidmm , remd^ Um. 
mebnnt omnid reCfdngnU , qnomm vnnm ^ e i I , ftne omnid re^ 
jSfdngn/d conttntdfifk ab enmftngnlis/nisfdrtiinsy^reiiqnit 
fdrxiins yfnifi/qnidlterdomninm qnddrdtornm^ qndftuntidi^ 
midqsy fine totsdemqnddhfUwrnmtotinx ib^Snodfiterdtoften* 
dtmdsan^e^ 



p 



rrofofitfo XIL 

Arabola fefqmtertia eft triangiili.^eandein ipfi bafim » & ei 
dem altitudinem habenti6 . 




Efto parabola ABC cui» 
diatneter BE,&€iPca parabola 
(ii paralleiogrammum DC •Du- 
caturqua^iibet FG diametro pa 
rallela ;eritq; F G • ad G I , vt 
B E ad G ! , fiue vt ret^iangulUm 

C E A» ad CG A»hoc eft vt quadratum CE ad redangulu 
C G A • Et hoc modo femper > Suntq; prims ma^tudines 
femper fqualesr q&x B E ;^ tertia? autem femper «quales qaa<' 
drato CE.. •Ergoomne5prrma?limul,'hoceftparallek>granv £^,||. 
mum AB^adomnes^cundasiimuI, nempead fenniparabola 
AI BE ; erunt vt omnes finfiuI^teFtias , videlicet totquadrata li- 
tiea? C£ quotiprahabetptffes-y^ad-emnesquaftasfirntiMiem 
peadomniare<5(anguIafuD C£ cumiingdfislftuspaFtibas, 8c 
fub reliquis partibus • £rgo ( ex prdtcedenti lemmate ) paralle^ 
iogrammum AB pritipfius fecraparabolf fefquiattetum : To* 

mmq; 




60 Be DimenfioQe 

tuinqucparallelogranunum DC crictotiusparabol«ic 
tenim » nempc vt #*• ad 4* Proptereapaiaboia ad inicripcimi & 
bi tt-iangulum /"qqod quidem paraliclograsr.mi DC fubd»* 
plum eft ) erit vt ^.ad 3» Nempe fefquitertia ^ Quod ecac fi«. 



ft fdtm 4ifg$m$em$0 ^ dimafis immem frimchfs , mem^e ferfmf'^ 
fofitUmemfrepertUms , qudm tflimdrus idbei ddffkfrmm^ 
imferifidm \ quu quidemfrofertiefefnuUUerd efi^ vt ^Semditm 
#jr Arehimtede ; Uirro Prim$o de 5fkfr4(^ CjUmdre » 



Prefofiih XIII ^ 



p 



Arabola fefquicertia eft trianguli eandem ipfi bafim, & ea- 
dem aititudinem habentis » 




Edoparabofa ABC» circa 
quam fit paralleiogrammu AD > 
&circa diametrum AC fiat fe- 
micirculus , circa quem fit redan 
gulum AE« Tummanente axe 
A C > intelligamr circumucrti ip- 
fum femicirculum , ita vt ex ipfi- 
us reuotutione Sphn^a circum-^ 

faibanir:exc6uerfionever6 rcAang.AEcylindrus naicatur. 
Sumpto iam quolibet pun^o G» ducatur reda G F paralle- 

la diamenx) H B }& per idem pundum G agamr planum G L ere 
dumadaxemAC 

Eritttda FG ad GI > vt BH ad GI ^ ob «{iialitatem) 
hoceftvtre^aogttlum C H A,ad redangulum CG A,fiue vt 
miadratum H N ad ouadratum G M /^ob circulnm^ fiue vt qua 
dracum GL adquadratum GM; nempe vt circuiusex femi* 
diameoro G L in cylindro^ad circuhim ex femidiametro G M 
in fphasra • £t hoc f emper^ vbicunque fumatur pundmi G,5um 
autemaequalctinterietamomBes primxt quam Mnnes terti« 

magni 



Problema Primiim« 6^ 

magnitiidine5« Ergdomnesprima^» netnpeparallelflgrtnimG x 
>KD adomnesfecundas^ nempe adparabolam ABC, erunt 
vt omnes tertur > hoc eft cylindrus , ad omnes (imul quiFtas» vi^ • 
^lelicet ad fpliao^am • Sed cyliodrus ad fphaeraln eft teTquialter ; 
crgo paralleiogrammum etiam A D parabolae fefquiakerum 
crit: &ipfa parabola iofcripti fibi trianguliiefquitertia s vtin 
praecedenti conclufum eft • Qupd &c. 

Limmd XXII. 
Si magnitudints quotcunque ad libram appenfar fuerint ex 
quibufcunq>pundis: cotidemq; magnitudines alterius ordinis 
cxijfdempunaispendeantiparitercum prafdidis magnitudi* 
nibus proportionales • Eritvaumideiiiq; libras^pundum cen- 
truma^uilibrij vtriusque ordinis magnitudinum • 

SmtadUbrjm ab mAgmtMdi . 
nts frimiifrdinis qMtcmnqut c, . ^ 

d,c,f, tx quibufcunqut fun- A^ T)L '^^X^^J^ 
HisMfftnff .Totidemqutmdgni ^J jfl C^^^^^j^ \y 
sudifus gyh ^i^\y fecundi^di- * * 7 

nis f€nde4nt £X qfdem fun£tis ; 

(^fintffOfmi^ndUs : nemft :Ft c 4ddy itdfit g W h ^ //r* 
eumut c dd e~, itdfit g^d i. g^. Dic^ idemfmtStttm Mru 
ejfectntrum cmmu ne 4tqu$likrij tstrimfque i^dinis mngtskndU 
numfuffenfdrum. 

Cumeuimfitutc udd^ itd g udhyexeodem fumSfi fft^n^ 
dtrdtuntytdmduf mdgnitudinfs c <^ d> quumduf g ef h« ' 

AmfUhs^ CumfitMt cMlAitd^ dd h» trit €$ nmerfeud $ 
^ cemfenemUAc dd c y ut h% dd .^ . c dutem dd t tft mt ^^ 
ddii ergh ^ ffuecdfimulddcy erit ut ^h fimsddd i.^mu^ 
rtmdgnitudines cd, c^ e , ex eedemfum^^ fquifemde fdhnnf 
exqu0fqmf0nderdntduf ohy(^i. 

ylteriue. CummttemferidmMffdjfitutc4 ud^^itd^h 
dd ijdritcpmfttesfde cd^ dd t% mt ghidd f. Sed^ ddce^ 
middgi&Cddf.Mi^ddl.J^S^efitfquetdpfitmdiddi^ 

erit 




Z>ic0 tTiline$im mixtum a b c d fquifonde 
W0e exfunHo tdngentis cd^uSi edfi$ du 
uidittiftutfM^suerfuscontsUum c, reli^ 
fUdsittrifU. 

Cencifidturfgurd itd utd^ dJJiorizon 
tentfitferfendicuiaris\ fj^chrcddiametru 
d a intelUgdtur circulusy quifit bdfis cetti 
mvticemhdbentisinfun&o c« 

Sttmfto i^m qtt^hbet fttnSfo e ducdtur 
e f fquidiftdns iffi d a i &fer iffdm trdtrfedt fldnumfdrdUe* 
Istmbdficoni. 
^^'^ Efit ergore/Efd Azddthj ut quddrdtum Ac ddca siue 
^* mtquddrdtum d^ dd cf, boc efiutcirculus dadd ef. Et 
hcfemfertuticunqifitfun^um e« Ergo cum dd lihdm d c 
fendedntdb qfdemfuniiis mdgnitudines duorumordinttm fro* 
f^rtiottdles ut infrdecedenti lemmdte imferdtum eft , hdbebunt 
tmitesfndgtUtudinesfimulfrimiordinis (hoc e3 omnes linedt 
orilinei a b c d yfiue iffum trilineum) idemfun^um dequilitrqj 
qtiodhiAentomnesmdgttitudinessimulfecuttdi ofditus {hoceft 
umttes circuli coni a cd , siue idem contts. ) Cottus dutem dequi^ 
fOttd«tSiexf$ttoSh quodfgcdi cd itd utfdrs ddc reluptdesii 

irifld 



6% De Dimeiiiioiie Parabdas 

eritut ghifimmldd l.^Ergo duf mdgnitudines cdc» ^ f^ 
hdieiuntidemfun£futn dquslihfyquod hdient duf tndgtoiistM- 
nes g hi i^l^ Et sic etidm si sint flures mdgtumdmes %effjtte 
ininfittitmmf^uoderdt frofofittnn&c. 

Lemmd XX III^ 1 

Si paral>oIa tangentem habuerit ad bafim, ex akera vero par 
teiineamdiainetroparallelam .Trilineum comprsehenfum iub 
curua parabolica r fub cafigente9& fub paraliela pr a^i&a>aequi» 
poiid€rabitexpuii€totangeDtis vbiea fic diuidimr, vt parsad 
conta^um terminata reJiqua? fit tripia • 

MBofdrdhld abc» ettius tdngens dd 
hdfimfit cd ifquidiftdnsdidmetrofit ad 



Problema Primum • 6} 

tr$pU,qitMd0qMidemread^3^ tfi*d hwi^otttem fetftndicw 
Urisierg» etidm triUneitm abcd dquifitnder^bit ex eedem 
fttn^», ^oderMfrofoaittim&f' 



Frof^fiiio XIV. 



P 



Arabola fefquiteitiaeft trianguUcandemipfibafinfc 
dem alcitud incm habentis • 




Eftoparabola ABC, cuius diameter 
DE intelligaturadhorizontempcrpendi- 
cularisifintqueCF, &AD tangentesj 
ipfavcro Af diametraaeqtiidiftans . 

Sumatur deinde F H quarta pars totius 
FCj&cxpun<ao H (pcrLcmma praece- 
dens^ a?quiponderabit orilincum mixtum 
ABCF. AccipiaturetiamFItertiapars 
totius F Cy&ex I a?quiponderabittotum 
tnanguHki AFC. Parabolavero,cum 
habeatcennrumindiamctro, aequiponde- 
ratexD- Ergotrilineum ABCF ad ipfam parabolam erit 
reciprocevt DI ad I H, pempe duplum rqualium enim par- 
tium FC aft i2;taliumif^fa FD eftif. FI vero 4.& FH j. 
&ide6 DI 2, & IH vna.) Propterea componendo erit to- 
tum triangulum A F C , paraboJa? triplum • Reliquum quadra* 
tiir^ abfoluitur vt in Propofitione I X. fiidum eft • Quod erat 
&c» 

Aliter. . 
PmtisijfdemyVtfi^i^finmsim fh, qmdirupdrsmnis £c^ 
s^mifQnderdibitq; ex funita b triUneum mixtttm a b c f . Sm- 
mdtmrtndim fi , tertiafarsipsims fd \tunc enim dtquifanderd^ 
bit ex funifo i triangmlum f d a . Trilineum uero mixtum a b 
c d , dqmifonderdt e%fmitSto ,d -. fudtm triAngulum totum a d c 
mquifondermt exfun^o difdrtlkeUeti/imahUtdexeedMfun- 

I HoA 




64 De Dimenfibne Patabote 

00 d fqmf^nder^ erg^^dmfeUqnm 
mlineMrmzbcd expMMUd^iimifm- 
derdeenecejfeeft. ) Erii itdfne tnMm^ 
gnUm fda dd mUnenm abcd nt 
tecifreci d h ad h i nemfe vi i.dd 
nnnm; & ftr conuersionem rdiionis 
tridngmtnm a d c sdfdnAeUm eriimt 
3.ddi.simemf6Jul4. ^drefdrdAeU 
mdiridngmlmm a b c erit mt ^Udi . Ne^ 
fefefqmitertia. ^ederdifrtfesiimm 
demonjitdre . (^c. 

Alijs etidmfrincifijs fdrdboU cfmd^ 

. dratmrdm dggredidmmr^frdmifsdfeqmeniifngreffUmi 
metricdrmmffecmUtiene . 

Lemmd XXIV^ 
Si duae xt6at linese iimiccmconcurnint^ inter ipfas dclcr^ 
tnmficquoddamikxilinainconftanscxlineis altemadm pa- 
rallelis ; eruncoovies linea^^qua^intcr fcparBHdxiiBii^ iaa}ii* 
tinua proportione • 

Cencmrrdnt inmicem dmde rt. 
ffdlinefj ab, cb infmnctt b; 
C^ inter iffds defiriftmm sitflexi 
Unemm czdefg.&dtdmt ca» 
d e > f g ^c. sisumtirft fdtdUe^ 
Ifiitem ad, ef, e^reliqmde mi^ 

cisimfmmftdeinterfefdrdlleUesint^ 2>#>#ac>ed, gf» ^e 
in centinmdfrcfortione « 

s\xti . dh dd hf, Uc e^ cd dd ^L fenB^ drgo qmod frffositmm 
fmadt^ 




i^ % ^ 



Lemmd XXV^ 
Pofitisdiiabus rcais lioei^ imiiccm^xincium 

Siioter 




Problema Primum # gg 

Cinteripfasfiierintduaeparallelse AC^DE» Scimiti CDb 
continuatamintelligatur flexilineiun ACDEin infinicumvf^ 
que ad pun^him concurfus B« Dicoinhuiufmodi flcxilkieo 
cflc omnes>& fingulos ad vnguem termmos qui funt in progrei« 
fione proportionis A C ad D E • in iflfiRmimcontinaataet 

Tondtnr £ fqndlis iffi zc ^ & 
g dquAlisiffi d c : Etc0»cifidtw 
fr^prtto f ddo continttdid in tn* 
fmttsfmstermims f H* 

Idm^fifcffAiU efiy dUqnem^ 

fine dli(fM4s termims effe infre^ 

grtj^ne t H,f « ne referidntnr 

in fiexilinee . EHo : &fit mdxi* 

mnsterminns i^iUernm, qnieiimfim mfregreffidne f H,n»m 

fnnt infiexiUnee . Eritergo tenmms I tffi frfcedensyinfit^ 

xilineo . SitiUe mn. Etqno nidm 1 nd i eBvt fdd ^jfinevt 

^cdddcfimvmm dd^pofrMm^fefnensem,fitntq;fqnd^ 

lesly & nmi ermu dfmdtejnmdmi & po. Termimts ergo i 

qnifontl^dtnr non efieinfiexikneo \ in oodem refertnsefi^ 

Eodemfetrimsmoelo demonfhdkmns nntinm termittnmeffi 
infiexiUneOy qnimnfit etidm in frogtofjtene f H . ef f • Concbo^ 
demns igitnr efe infiexilimo omsoes frfcisi terminosfrefeftid^ 
tris zc ddAc init^finitttm continndtf^ ctmt demonfirdtnm fit 
ttnUnm in fiexiUtuo termittnm defiderdri cfttifitim frogreffiomo 
f H > neq; vUnmfwferdtnnddre^ qni mn referidtnr etidm no 
f rogr ejjione fli.&e. ^ 

■ 

Lemmd XX VI. 

Su{^fitis infmitis redts lineis in contimia propbrdone maio 
risihaequalitaciStreAam lioeamtqusepraedi^omnilMis fitas 
qualisreperire* 

PondnUtrfritnddndlinifddtffrogrefionisefilt a» b / qttit. 
fosumtnrfiptddsi cd tmtioriz^&tftnitmi hJintq; cdf ef 

I M fdsd^ 



66 De Dimenfione P^rabolg 

f^MleU } & iungMtw d f , c c , f$i( neceff^riocom^urrtntXX 
cnn4ntii^;inpnna0 g, &dMCii QUiffi^qutdifidMsJit gL 

Bic^o re£f^m d 1 kqudem ejfe 
infinitis terminis frogref- 
fionif a b m fimuLfumftis . . 

Concifidtur enim continndtn 
fiexiliheum d c f e &c. in infini- 
tum, vfq; ddfunolum g , eruntq; 
in iffo omn^s iinef , fiue termi- 
ni ddtf frogrefftonis a b m^ 

Produtdntttr idm he, ni. & 

reUqUMiffisfardUelf^fq;4d AL 

^syw- £,,y^y^ g f ^ fqudUs iffi cp,&h 

j dqudlis iffi p C[ ; & iioiffi<\U 

&fic4efingulis . J^Udliktt emim 

linedqudfitinjUxilintoJ^^Mt 

fudm fortiunculdm reffotodtntem 

Utrt&d^\iJibifqudUm;d$necfUxilineumftrutnfrit udvU 
timumfunifumgx TuncduttmneqttddefiexilineOyH^quedt 
l^ned d 1 quUqudmfufererit ;fedtdmiffum fiexilineumy qudm 
tti^mrjtSid dl finitusdtfumftdntitz EB enimiffdgly q0d 
jA vltim^fiixiUneifuniU g ducitur , vltimdommum fnrdlie' 
i^rum y qud frodncuntur vfque ddd\. Ergo omnesfimul lined 
fiexilineij qturumfrimd eft cA^ dlterndtimfumftd (bac efiom. 
^s lintdd ffrogteQunis a b m)d€qudUs funtotimib.fortiunctdis 
mtifdt 41 ytmmjumftis :boc efiifsi dl . ^u/tdetstoSenden^ 
dttm&Cj, 




lemmd XKVII. 
. Suppofitis infiflitis magnitudinibus in condnua proportione 
Geometrica nkuoris in$qualitatis# erit prima magnitiido mcdia 
proportionalis interprimam di£fcrentiam & interaggregatum 



ommum^ 



Jlfiumftienim ftdctdcBti ^tn/ku^itne^ ditcdturlvL /tquim 

difidns 



f 

I 



^H4nsipsi g c ; c^ eriP d u pri- 
fka difftrentid . SedAwddfTi- 
mum magpiitudinem. dcrlfutfd 
4^ d g , hc ejTut dzadd^\ dg- 
gregdtum omnium. ^upd erat 
demonHrandum &c^ * i . 

SCHOLIVM; 



^. 




/•v 



Hoc efs^e uerum etiam in nume 
^i^ > cfc cuisefcun^, g^^^^f^ magriitudinH^us non duhitakfmus af^ 
frmdtt^ 4ffenemus etiam umuerfalhr^m demonftrati0pim^\ 
ffaecifuecumadmodumtreuissit . ffuius ueritatis conclsuio^ 
cttm a nobis obiter celebirrimo Caftaleriocollatafuiffet^ ipfe eiUb 
idem Theoremafequentidemonftration^^quf anobisiam^injki^ 
mainutntione adhiiitafiieratii/onfifmauit . 

Frfmittititr4joc i J^od sifuerinf quotcunq.mitgnktidmasi 
siuefnitf.numero , siueinfnitf^ quarum ant^c^densfemferfe*: 
ijuentemaiorsityttitfrima omniummagnitudo fqualis omuib» 
different^s sifnulcumifjamiriimk mdgniiudine fum^siyv : 

NotttmefhocapttdG^omttrasydemonftratstrq.ut a mobisfa^' 
Itumeftinlemmate ij. Vbio^endimus faraUeloffratfmutm^t> 
fqualeeJfeomnibusdiferentisinterfequentiafarMeUgramma^ 
minimoparaUeUfframmo oc. 

Suffonanturiam infnitaenumero magnitudinesin cantinua. 
frofortione G-eomctticdmaiorisS inaequaUiatis ^ fndnifejtumaft ; 
quod minimaimmum mMgnitudoval n6n ^rit ^ ueifuncfttm>em^> 
Ergo in hoccaftt eritfrtina magnitudo aequalis omnibus tdntum 
differentifs. 

Cum auten^fondnturt^agnitiulines in c^ntinua frvfwtiona. 
Geometrica , rruntttidm diffeirentia in eadem rationefrofortio-» 
haleSyC^ ideo (faota conuerjhnejerit ^vtfrima diffirentia adfri^ 
mammagnittidinem^ itafectfnda differentia adfecundam ma^ 
gnitudinem , ^fiifimfer^ . PP4fterea4^t vnd advnam , itacol^ 
le£tim erunt omnes ad omnes . Nemft vt frimd Mjjhemia ad 

fri^ 



4« ScMti. 



..1.^1 



■ ' • 



Si De Dim^fione Parabote 

frimdmmdgmiiMdimfm 9. itd irmmiomm€sfimMldiffercmiut{h$€ 
ejl iffdffitmd mdgmitmda) dd^mmes mdgmiiMMmesJlmnd . Cem* 
ffdt ergoffimuim mdgmitttdimtm medidmfrsfmismdUm efit 
iirfrimdm diffitnttidm^ & dggTigdttatnmtttittm ^ 



p 



Profofttio XF. 

ArabokfeicpiiteitoeftcrianguUeaii^ bafiin»2c 

dem altitudioem habencis » 




Eftoparabola ABC inqua 
infcripMtn fit triangulum ABC«^ 
Dicopdrabolamtrianguli ABC 
edefefquitertiam» 

Inicribantur enim etiam in re- 
liquis portionibus ADBfBECtduotrianguIa ADB^BEC 
Edtqi trJai^;ulum A B C quadruplum duorum fiuuil triaogiilo- 
rufp A D B> B E C« Cocicipiantur etiam. inreliquis quatuor 
poreitiiicuUs AD» DB» B£,EC#iafcripcaquatuor criaagu- 
Ia;eruntqiduofimuItriangula ADB, BEC quadrupk prae* 
diftorum (imulquamor iiit^quentium triangnlonvu; &hoc 
modoietnper. ParaboIaigiofirQihilaliudeft quam aggrcga* 
tumquoddamiiiiimtarinnnuaieronugnitudinamin proportio 
nequadrupkyquarum prinu eft nriangulum ABC> fecunda 
veroconftat ex duobus triangulis ADB^ BEC. Propcerea 
piimamagttitudo ABC medxaproportioiialiseritintcrpriaia 
ditfcrentiam » & aggr.egatum omniumt nempe parabolam . 

Po«aturicaq;oianguIum ABC eiTc vt4»&;ideo duo (imul 
triangula A D B» B fi C erunt vt vnum : ericq; prinia ditferen- 
tia/niinirumtnter4«& vnmn^vtj. Ergo aggregamm omni- 
um in&ucarum magnitudinum » nempe ipfa porabpla» erit /"per 
lemma a 7. ) ad primam magnicudiaem^hpc eft ad ini^iptu tor 
angukim ABC»vtprimaipfamaenimdoadprimamdi&reR- 
tiam ; videlicet vt 4. ad 3^ nempc icfquiteniair Quod eratpro- 
pofitum demonftrare && 

Aliter. 



* • "■ * . r a^ 

Problema Pdmum « 69 



Aliter. 



♦, 




ter d b, tangeniesddbdjim a d , c d, 

^erverticemvero t(. InfcrihMtttr 
juitem in reliijuis trilineis a b e , b c 
f , duetridnguU g e h , i fl , (vtim^ 

.perdtumfiiitfro confiruBione tem- 
fndtumTertyy(^ ^arti.) Itemin 
reUquis qudtuortrilineismixtisMUd^ 

tu§rtridnguUconcifidnturi (^ooc modofemper. JEritf; vni* 
Merfum trilineum a b c d nihil dliud qudtm ^ggi^^gdtum quoddi 
injlnitdtummultitudine mdgnitudinuminfrofortiont quddrn* cani. «, 
fU^ qudrum/rimd eft tridnz^lum c d f ^fecundd^verh confidt ex Lem.i. 
jduobus tridngulis g e h , i H ; tertidvero ex quditiinr fequenti* 
hus^c. Propjeredjtggregdtumomnium^nemfetrilinettmmix^ 
tum zhcAyjtdfrimdm mdgnitudinem j nemfe dd tridnguUm 
e d f , trit vtijfdfrimd mdgnitudo ddfnmdm differentidm^vi* LmM. 
jdtUcetvt4.ddj. 

Cumitdquetridintttm abcd ddtridnguUtm tAtjfitvt 4. 
ddtridy^ritidemtrilinetmddtridngtdum adc vt4.ddi2.(^ 
ideo fdTdhoU dd tridttgulnm adc eritvtf.ddi2.(!rddinfcrif. 
tum fihi tridngulum vti.ddg. Nemfefefquitenid . ^npderju 
dtmonfirdndttm drc^ 

Ztmmd X xrill^ 

SiiueiiRtinfmitatniiniero reAae Une^ ABy CD, HF/ftc; 
in cominua prc^nione Geometrica maioris inxqualitads i aK 
tera autem ponatur progreilib fi G> DH, F 1 5cc. ita vt fit opca!^ 
admodum ABprimaadfiG primam^itil CD fecundaad D 
Hfecundamt&itatertia EF adtectiam FI &(icfeitmer.Di« 
co vniuerfum aggregaram progreffionis A B, CD j E F , dccad 
^ggregatumpro^^onis BG^DHyPIiCflevt ABadfiO. 

IntiU 




iuxss £f 



h§m»i6» 



4» fexii 



D^ PimqificHie Parabolas 

intelligdMW emnes termini dud 
tum fTOgrt^num effe in flcxiU^ 
neis&c.iunSHfqi ad,gd»4^itf^^ 
t$ir oi pJtrdlleUiffi ad> (^ om - 
fdTdUcUiffi dg, Eritq-^ b 1 dqud- 
Us omnih: infinitis terminis 3.b, 
cd^cf^C^.iffdvero om dqnd 
Us,0mnibus infinitis terminisrc^ 
JHqud froffrejjtonis b g > d h » f i • 
' /^w ; vt Ib 4^ b a, /« efi o 
h adbdy hoc efi mb dd bg. 
Permutdndo i^tur , Aggregdtum 
Ib^ ^ dd dggregdrum bm, ly^o// 
ab ddb^\nemfevtvndmdgni^ 
iudoddvndm. ^tfiderdt &c. 

/r^^ Theoremd foterdtfuffoni tdmqudm demonfirdfttm in 
Profofitione Z2 .Ubri V. £uclidis: vnum enim dtq\ idem eH cum 
Thearemdte di£fd Profofitionis : Verum , quonidmfere 0mnes 
ofindntur EucUdem ibifuffonere muUitudinem mdgnitmdinum 
finitdm , voUimtts dttxilio fiexiUneorum vti, 

Profofifio XVI. 

Arabola ferquitertia eft trianguli eandem ipfi bafim, & ean 
dem altimdmem habcn tis * 




p 



Sitparabola ABC, cuiusdia- 
m^^tef D £ , tapgcntes ad ba(im A 
jE),GD:perverticemvero FBG. 
claDgulum infcriptum A B C, Di- 
co parabolam tt ianguli AB C • ef- 
fe lefquitertiam • 

a fata^ ^n^ e^i^ ipf^ E ^ asqualis fit 
hpUm. MBD> re^aa vero AC dupla A 




rc^ 



f Probteim Primum# 71 

redaeFGientinicripcumttiangiiiuin ABC duplinntrianga- 
I4 F D G fub tangentibus con^aehenfi • £t hoc ien^ier vefvni 
d9:etttmcircarel^ttasjK>rtkmesp^^ AIfi> BOCt 

^^enjm AIB parabolatcuiusiangentesadbafim func AF» 
BF, ideoq; trianguhim io£:ripium AIB duphun erit oian* 
guli t:^ngentium L F M • Idemq» verum etiam eft exaltera par-* 
ce ; Ergoduofimultriangula AIBt BOCt dupla funt duorii 
fimul L F M 9 N G P .^ ergo cum fint dua? progreffiones vtraq; 
ifeproportionecontinuata magnioidinum wfinkarimi mukttu- 
dine » ( altera nen^pe intraparabolam » cuius primus termimis 
cft viangulum A B Cji fecundus vero > duo uriangula fimui A I 
E» BO C 8cc. altera vero prdgrelfip extraparabolam» cuius 
nempcprimus terminus eft triangulum FD G ; fecundus au* 
tem duo fimul triangula L F M» vN.G P « &c> ) funtq; finguU tflr-- 
mini progreifionis i cpe intra parabolam eft » diipli finguiorum 

ternunonmiprogreiuonis>quasexir4eft;EricergoaggregaQim j^ _ 
vniuerfum prim^ progreflionis duplum todus aggregati iecun- * * 
dae progreflionis s Nempe ip(a parabola dupla eric orilinei mix- 
ti A B C D . Componendo igitur , & per conUcrfionem ratio- 
^nis^ erit triangulum A D C ipfius parabolf fefquialcerumsnein 
pe vt 6i ad 4* ideoq; parabola ad tri^gulum A B^G erit vt 4 • * 
ad 3^.videlicetfefquicertia« C^deratoftendendum&c. 

. rdrabolf^Mddrdtwd bdteri fottfifumftis dUfsfrincif^s^ope 
tdmen indimifibilium . Suffonimus^Uf Archimtdes demonfitd^ 
ttitif^librodeUnfisSfkmbusMdProfofitioneS' /^, i^zs.Pru^ 
miJ^JLotmdtebmo^mdi. . . 



f • « 



Lemmd XXIX0. 

Si fiicric vtprima magnitudoad fecundam»ita terdaad quar- 
Cam ft& hocquptiefcunqUibuerit : fiierincq; omnes prim^, item 
&omnesteiici$ eodetn ordine proportiopaie&:; Eruntomnes 
prima?fimuladomhesfecundas> vt funt omnes teru^fimulad 
onmes quartas • 



> 



K Sh 




ni-'i 



I • 



I. 



r 

e 




aI eI il 

B{ FI L1 

Rii sOtS 



D 



D 



gU 



2«. 



T^ Dc 

Sit a frimsad h/Mmdmm-y^ e 4mid 

^MiefrmHfiJihum . Simnfmesmmes/ntme 

fnfmhmdssex0rdmeiJKr>mifet^ ^4§d c 
uyiffcadg. Aii^li»sfvex sdwMdfe 
cidtti&e.etfiejemfer.' Dkemmtifri^ 
mMs fimml z^ c» 1« tie. sd em$mes feemmdds 
fimml b » f, 1 e/r. e^cvtfimfemaus tetiUfi^ 
stsml c^t^^vXyW.mdemmesfmmmms/hmml^ 

Actifimfme O , p> q ifimgtddmqmdesfri^ 
mtmfrimmnmmylfec eH iffi^i etfimtetulem 
(fmetftmtenmme/frmmdz^ e, iiett^Itemfm^ 
mumtttt r, f » t ; tetidemtptetfimt^^ottmester^ 
eifietfimfim^fv^itXsmftmlesfrim$fter 
' -fimnmmtsmfti^^:^ 

iMmei fqtmditMtmeritve o^ z^itdx 

mdc. Amjdius:€mm^fitnqmlisiffi^^ef{ifficyerh(ffe- 
fter fieffefieimem)vt fmdtjitdidd^.ethoc ftmsfer .fmm 
Um.\%^ ^meemmes o ,p ,i] ^ufttdes ^ isemmf^emkes r» f , t , p^mdlesy trp 
ermmtemmespmiU o, p ,q , etCddmmmes a , c, i , etc. mrmmmes 
r , f , t , fimmly ddommes c , g » m . pemiqme csmucruntU , em$- 
mts^jC^ifi^^^^mmesOyf^KlyermmtmtemmfesCy^ym^ ddmm^ 

^/tj,u ^mdm$mtemt. 

^e midm tm e t^m^h jTi^ si,>M r dd-c^^etm^ddbfftm cdd 

d .' er$t exdqm o ddhy mt r dd d :>Eddemfenitmrdtiemecm 

elmdemms exdqm efiemt p ddf, itd ( ddh: et fic de Cfterit . 

tdm.1%. Ermmterg0ommesfimid^\p^^y¥tt.ddemmes b, f»i» ete. m 

fimtemmesfimml r^t^X-^ 'etc.Jaiemmesd » h , t) ; etc. Sjtdre ex 

^qmermmttmmesJixyC^i^eec^pmmeshffy Utu.memtfescy 

^tm^^te.ddtmmtee^dyh^^^etc^ ^tfederdtmammderodmm 





HDN|| 



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nf^ 



Frob&maLFri 



• i MMI 



71 



l?TOfafak XV 11. 

T^ Arabolafc%attcm bafioitftci 



dcm 




iiabciitis» 






SttpaisabolaA 
BC>cuiustan« 

;ciis fk A E>. 

liamctro vcro 
asquidiftans fit 
C £> & duca» 
nir quarlibct F 
D» parallclaip 
(i C£> £iricq; 
ECadFBJon 
gitudinc .» vt £ 
A ad AF>fi* 
ucECadFD 
potcntia^Ptop- 
tcrcacnintinco 
tinua proportio 
nc. £C,FD> 
F& 

Fkntdcinde 

cennro A^ioteruailis AC» A D, duo^circuli; &ponatur£Ij- 
cis initium cx femidtametro A C^ 'Sitq;ipfa.clix A G C. 

Erititaq; DF ad FB^vtC£ ad DF ; fiucvt C A» ad A 
Dyhoceft vt CA ad ACiiuevtperiphfriatotaXLHQad M.>^f# 
arcum C LHrfaoceft vt pcr^h^ia tota D P G D, ad.arcum D **••• 
P G • Atque hoccritfcii^* viueuBqiie lumatur puniStum D ., 
Siintq;on(iBcs|iriinaK^ittinoip|iestertwe Qfiagnitudincs^eoQio^ 
doquadebcntproportjcnalcs^vtinlraaftcndemus.) Qiiarc 

omncsprinirfimtt),nenf^tnangulum AEC^adomnesfe- lemmm 
cundas &xkvi nen^ad n:ilfncuiii.inixtHm A BC £> crit f^^^^* 
ntoaiay^tense&x^^ CLflAad omnes quar. 

K a tas 




\ 




T4: ' DftBliianefifiL _ 

tas litnul, hoc left ac| reliquum ipfius circuii» dempro helicis ipa- 

tio C A G C • Circukis auscm C LH ,, di&i fpatij, dempto he- 

%sdi li^ licis l|>atio»fefquiaiter eft ; Erao etiam tnai^ulum A CB fcf- 

'sM^I!\ q^^lteru:nerit triliaeimixti ABCE3.£tpcrcoiiiiedioaeimrar 

* cionis f triangiilum. A C E , triplum erit parabol^ A B C • Re- 

liquumquadratur^eabfoluetur vtinp^Propofitiotieiadiim e& 

JljipddMttmdffkmptumfiiit , tmttc $fiendemus ;/ciUcetftM 
emkfrfrimf , omnefqMc ttrtutmdgt^tmdinesfimtffefi^rHenults 
€0 minfo y vt rtquiritwr in ItmmdttffMtdcnti . 

Ducdtur infrfmifikfigurd , qudUhtt m o » Mqmdifiuns iffi i 
d ; &fok4mus iffitm f d tjjifrimdmfrinusrum\ iffum nfcrofcri^ 
fbdrid dpg frimdmttrtidrum . Erit trgo diddomsWda dd 
a o 9 fint vtftrifhfrid d p g dd fcrifhfridm cstiusfcmidiumucr 
dfiao drc. Etjicftmftr . Sluod ofmchdt drc 

PdTdtoldm ttidmquddrdl^imus inttntdtd ddhttc vid r nimiru 

qudfito tius ctntrogrduitdtis dfricri ofc ituUuifihilium . Suf^ 

fonimus duttm Ipnmdy qpod Archimtdcs oHtncUt inftcundc 

jttquifondtrdntinm. Hoc tfifdrdholdrumccntrdgrduitdiisjm 

cddtm frofortiontfuds dtdmttrouftedrc •. 

». 

Lcmmd JTXX. 

Centrumgrauitatisparaboiaediametrumita diuidit» vtpars 
ad verticenv terminata yreliquas £t f efquiaiceca . 

Efio conusquiRbtr ^h Cy cuiushdfis amc,iAMr bd , trUU 
guluni vtrofir dxtmfit a b c ; c^ (tBusfit contttfldnc cfg^vt 
iuhtturinXI. Propofitiont&h.frimiConiccrttm. Eritqueft- 
{iio qud vocdttirfdr^oldj iUiufquc didmtttr crit fh » Mfio ium 
crnirum grduitdtisfdrdhold efg, quoJbtir pttstBstm ^ futd L 
00tndtndumtfirtCfdm{i ftfquidlttrdmefio ipfittsih. 
' Agdturftrfunilum i rtS^d aii iftcttmqsu cottus dU^ffld^ 
ttomiiOr ifp t f^fdtdUr critq.fcBio m a o fdrdholdy ^ emr 
centrumgmmtdns crit ^ (^fi wim ^fdrdUcUc^ vut f i udi h| 



Pcobleixur 



75 



itdhpddpr iftd ipomtw c?* 

trum grmiutis f4rdMp e f g ; 

€rg0jf§rPr0pofit. 7 . Jii/ifMmii' 

4tqM^0»d0rd»tim» pcemrmtB^ 

ffMitdtis erit fdrdMf m n o*) 

Mtfieftniftry "ubiennq.fitfls- 

mnm m n o • Ommmm ergofin^ 

gilUtim fdr^Urtm$ fMf/MMt 

in €on0 a b c > centrdgrdttitdtis 

TtferiuntttrinreSd al *- ^uro 

otidmcommnne centmmgrdni 

tdtis omninm tdrnmdomjimnli 

frfdiifnrnmfdrdtoUmm crit inrti^da I • Omnesdntem fdrd^ 

k$lfydtq.iffeconns'tdemfiintiwrgocentrumconieJl inreSfd a 

1 iquod ctm$fit etidm iudxe b d ^ eritcentrum coni in cQmmuni 

concurfu i , ideoq. b f erit iffius^ f d trifU . 

Ducdturex centro tdfisreifd d q , fquidifidnsiffi z,h etunt 
fUCfqudles cq^ql « Ci$m dutem obcentrum contiffd hstri^ 
fUfitiffiusi^^^t etimnhl trifUiffius^\<\x^ ideohlfef 
quidlterdiffius 1 c: J^udre etium iifefquidUndetitifsius i h» 
^iiod erdtfrof§sitiimf^Cr, 







ri. 



TrofttfitW' XVHL 



P 



Arabola fefqukerda eft trianguli eandcm ipfi bafiav & 
dem aldmdinem habentir« 



- Eftoparabola ABC^ cuiiu> 
dtameterBD: infcripmm verd 
ttiangulum A B G • Dico para^ 
bolamfefquitertiam dit triangu 
l^ABC. 

Secentur bi^u-iam A D , DC 
jripundis £»& F:duda?q. £G> ^. £ 
I^H^dianKCroapquidifbmtes»^ . 




f?dio- 



't 



7^ De DiitteiifibiieParidn^ 

la^diametricruncportionu AGB, BHC. Siniamra graoir- 

l^^fft tatis di^aarum pordonum O^&Nieruntq; vtrag. GO, H N, 

c.^<«/, fefquialterareliquarOI.NL. lungatiirON^Arinipik ON 

%. frimi eritcehtnun commune grauitatis ckmii]» jMMfonuin : fcd cft 

^^«'f « e tiam ia B D fnam in B D eft tamccnaCDiltcmus parabols^qoa 

etiam trianguli A B^Cj Qu^e pundum R» cemnim erit porcio- 

num AGB,BHC . Ponatur BD partium ^o.eritqiic GE» 

^cum (it iubfeiquitertia jpfius fi D ) partium 4 5« ipfa I £ 3 o. & 

ipfa £ O > hoc eft D P. 3 tf . Sit Q^centnim grauitatis trioQgoli 

lemfu ABC« Eritq. DQ^. 20. Sit R cemrtmiparabQl^ entq. RI> 

*^^^ 2^ Eritergo PH^ 12. & R<ii4. Sed vt PR ad RQjjarc- 

ciproce triangulum A BC adduasj)0]fiQnes AGB» BH C. 

Qtiare triangulum A B C ad doas pomones A G B , B^H C ciir 

vf i2.ad4. nempe vi^. advnnm; Coniponend^qiic»&per 

cOnuerfioncm rationis » erit parabola A B C ad in£7qicum €bi 

triangulum vt 4« ad 3« Nempcferquitertia* Qgsdcratpropo» 

£kum&c« 

ti404 Mdl^e itdtiikk.i^ddtMmdm fMA$U umddtmmfiim^ 
f*$pf0tnfi temmMte ^*qM$dquidem ISlcMd CMMMkriMmsfrmlijf 
fe rdMtum eH . IuferttiebMt.emmmemfiftf ttuufdMmfittdi 4A if 
fMfotMbdM citcaordinMtim:MjfUcM$Mm rettdutM^gemii . EH mm^ 
tem lemmM huiupmedi^ Audiorc lo JVntoot0 Roccha praeftanti 
Geometra » 

Zemmd XXJtl^ ' 
Sifigura plana (uper aliqua fui redalineafiguramipiamfe* 
cante libretur » erunt tpomenta fc^mentarcun figurae^ vf funt fo» ' 
lida rotunda ab ipfis fegmenti^ arca&cantemlineam ttsxAxs^ 
fts,dcicripta. 

EjtofigurMjiMUMifUMUtet a c d h it^quMmfecet teSuUmM 
a b : cf ceucifiMtut^mMlArMtifufetr^} a b . Dk^mtutn^ 
tumftgmenti a c d b yj^dmmttmhmtfiffmemti a:C f b.. egevtfi^ 
hdtmtttiuttdumgntUttmeKteutltt^mufifftietuixzAh citot 

dxem 



T7 




£roUtfi»PriiU)8i 

dxem a b » ddfeUimm r^mndMmgemium 

0X€9nHnri$9€f€liqiiif4f^€n^4iH^Kf$\ \ 
d€m dx€mf,€Molnii . 

Snfnfiiff mmdm i m f^m^ififit^ 
Bis \iyfir i * inreHd ab/ dncjimmf» 
h c^/rr i, f€&n cCtdf . f€ff€ndi€nU* 
T€s ddiffnn^ b ip€€nfnrf^ Sifigfdmf^* 
menSddU^ hffinfnnifisiy^m^ 

HdteHt €fg€ fncmentism reff^ dh 4d 
9n0m€nttnnr€Cld€ hf> rdthnan ^mtfdr^ 
sisdmexrdsionemdgmtndittnm dh dd 
hf» ^ €xrdii$n€ diHdstsidftim Ih ^ 
hm; sin€ dh dd \\i. Pr$ft€f€4mmmtnmr€&d dh ddtm^ 
m€ntnm hf €rit vt qnddrdinm dh dd ptddfdtMm hf. 

Eod€m modo oHendetnr momtntstm reffdfi c i » ddmomentnm 
reCtde ityefevt qnddrdsnm ci ddqMfdfdtttm i^i& tUffetto^ 

fefm . y 

^/«i^iErWffMii9M/irinr dh ddmitmltmftan cx^tfiijtJk.tdttdtm 
rdtionem vtfnfrd) vt qnddrdsnm d b ddqnddr^stnn c i : &itoc 
femfer^ ErMntergoomnesfrtttutcsimidmdgtutttdinds^ttemf€ 
onmidmomentdfgurde ac d b> dd omnesfecttttdds sittmUnom-^ 
fe ddomtndmomentd r eliquf figttrf a c tl> zvtfkstidwtmtstertif 
JimnL nempe omnid quddfdtdfignrde a C d b > dd omnid qmtdfjU 
tA reliqnde figttrdt . Sino vt fimttnmes circnit figmdt ac d b 
^n€tnfefoUA$mrptmdkm€xifstttscotsti€rsi$n€cmtLdX€m ab 
defcfiftnm) ddomnes circniosreliqndefigttrde ;x e i^{tt€tt^€dd 
fotidsfmrotnntt/im ex ifsins renokttiottt circd cntuhmMtm a b^ 
genitnm.) ^oderdt oSettd e ttt ltt m &c. 

H9efirfwtfff»,(fM§44iuuttvfifMfifJttJixim»sf€mtm jk 44* 
dtfimtftum <B , & hic igfertum t ^m/i m m JtUtmiim * ntfne. pid 
tgtfctam adhuc VMlgAttm )fdr/lboUm qM4drMt'tmtts iftfffffitit 
dtmpa^mittfe» umr mnhis nwiufffkstMr CjtUndrwm^nifti 
/iki< ( tm »i diijf4r4t0kei efft ditfhim, . 



Proft' 



V78 De J3llio^(k£oBk ^FuaboI^ 



p 



Pnf&fiii6XIX. 

VeinalritudmcnrifirbeKis.* . •• 




Efto femiparabola i^l^CD» 
drca quam iSt re^ngulum DE # 
Sumatur punt^m F , i&a vt A F % 
ad F D fit vt 5.ad 3. dii^aq. F<i 

Lem.tt. diametro d?quidiftans vcrit in ip- 
fa F G centrum grauitatis remi- 
parabols. Efto iilud pundhim 
quodliber» puta I > de par I duca 
<ur LIM parallelaad ADfac- 
cipi^turq; I N . aKjoaKsipfi I M. 

Inteiligatur etiam produda P Q, parallela diametro C D ^vbi- 
«Jnqvcadat) ka vtparaUdogrammum redangulum D P»a*qua- 
lefit ipfi femiparabolse • Tum concipiatur applicatum ad reda 

• « C D , redanguium D R , ita vt sequiponderet femiparabolx fa 
^iibratione fuperTcfta CD. Sitq;centrumdidiredangU'- 
lipundum S;&dudd TS X parallelaipfi A D iungaturreda 
IS. ^ 

' lam^iiiaaifeftumeft-ex lemmate praemitfo quod cylindius 
£i^nsare<ftanguIo DR circa axem DC reuoluto, asqualis 
^eritconoidaliparabolfeo^dodconuerfionefemiparabolse A 
CD^circa^eimdemaxem CD reuolut^ ; cumasqualia fuppo- 
fiantur figurarum pianarummomenta» Erit ergo cylindrus d re- 
danmilo D R h&us » fubduplus cy lindri d redangulo D E fa« 
€dj «ideaquadratum TX fubduphim erirqua(£ati ML^cy- 
kndri-enim asqueald&nt inter fe vt bafium qiudroto; quod me- 
Boento* 

4. /mi. yerikn MN ad TX,eft vt IM ad TS f^funtenimfubdu* 
pl« earumdem^ fiue vt I V ad V S ; ncmpe (qttia^^aequipofide- 
rantfiguraeplanaefuperlinea CD,fiueexpun^V) vtredan 

gulum 



Problema Primum # Sx 

gulutn D R ad fcmiparabolam reciproc^ fiue ad rcdanguiutn 
D P * ipfi femiparabolae xquale : fiue vteorum bafes T X ad M 
O • Ergo T X media proporoonalis eft imcr M>(» MO : Qua- 
re re Aaogulum NMO»cum«qu0lofiU)uadratoTXi fubdu^ 
plum erit quadrati L M • 

Ratio ver6 quadrati L M » ad Kdangulum K M O , compo^ 
Ritur exratione L M ad M N (quai ieiiqpiitertia eft per conftni- 
dionemj fumpfimusenimpun^^Fvitavt AF ad FD» eflec 
vt 5.ad 30 & ex ratione L M ad M O ; quxquidem ignota erat» 
fed necelfario fefquialtera nuRcapparct . Ratio emm dupla c5« 
ponitur cx feiqiittet«a^ & fefquialtera , vt ipfis etiam Cantori^ 
bos vulgatum eft i vt viderreft in hisiribus numeris 4. j, 2. 

Re^ftanguhim ergo D Eadipfota D P > fiue ad femiparabo* 
lam^fdiwsakeMmcrirjdiipfaibnparabd^ ad triangulum A 
CD.fel^ieraa^m. Qi»d«ncoftendendum&:c. 



Sitparabola A BC# wMsbtfis AC^ 
tangens C D ; diandetro aequidiftans fit A 
D. Sumpto quolibeit pqn^^ S« dtffatar 
£ F diametro sequidiftaos • Dico efTe vt 
FEad £fi,ita C Aad A£. 

£ftenim DAadFB IongttaBtiae,vt 
DCad CFpotentia,fiuevtDAadF£ 
potentia i Sunt ergo in continua radone D 
A > F£ y F B • Quod memento • 

lamvt AC ad C£ itacft AD ad£F, 
fiue £ F ad F B ; & per conuerfionem ra- 
tionis.vt C A ad Afi,itaeft F£ad £B» Qspd csatofta»*' 
dcnd«&c« 







Lemmd XXXllt. 

Qiuelibet parabola sequalis eft duabus parabolis fimul fuo»» 
ptis,qusequidem9qvaicmipfibafimhabeaat» diametrum ve« 

L r6&ib« 




$if DeDimenfioneParaboIa? 

ro iubdi^laiii , & arquaticerinclioatam •' 

tPffgrahoU abc, CMiusdiA' 
me/rr b h vjintq. duf alU fotd- 
boU acc. agc. i» tidembdfi, 
DUmttri "vero h e i h g , vtrdq. 
fubdttfUfif dUmetri h b : fed f* 
tiHdUter adbdfimineiinntMy JDi- 
(0 paraholam ab.c: d^ndUm ejfit 
figm-daec^.^ 

Sumdtitr emm qt/cdlibet fn/h' 
^itmi^bafizf:\&fitxtiidu3dq»ie ^van dquidifidtitt tU did. 
metrftmh\\. £rit bh dd nmtM re£fd»^Um ahc, dd re- 
HdngnUm Amc ifimevtre^d he dd mo. Etfermutdndo vt 
bh A^hc» itderit nm ddmo. ^s^e nm dttfU eritiffius 
mo. Eedemfetiitusmeda tfiendettir nm dufUeti^miffiits 
jnp, Ergototanmd^udlisefiiffi o^. Ethocfemfer . Prtf- 
tereaomnesfimuUinepfi^miA ,a b Ct.(nemfe iffaforato U a b c/ 
aquales crunt omnibus (imul lineisfigura a e c g , ( isetnfe dua- 
tusfarahlis a c c , a g c,'^ ^od erat ^e. . 

Trofofim XX, 

PArakoIa lefquit&itia eft trianguU caadcmipfi bafim, & ei 
dem altitudinem habentis . 



Efto parabola A 
B C> cuiusdiame- 
ter B E concipiatur 
ad horieonrcm per- 
wmdiculans^r&lp- 
fa parabola inucrfa 
ftacuatur . Produca- 
turCAinD,itavt 
aEquales fiat-.C A, 
AD>&&DCUt>- 




Problema Primum • Sj 

bn tcuiusfulcrumeft A. Ducatur CF tangens pml^oIaQii 
& A F diametro E B ^uidiftaos • Ponatur ctiam G H asqux» 
lis ipfi A C^ & diuifa bifariam G H in I » fit vtraq. IL^ I M, fiib 
duplare^^ £B.&:aequaliteradbaiimlnciinata vteft'ip£i £B 
ad AC. Fiantq;duasparabolar GLH» GMHfquasrpcrieai 
ma pra?ced.^ ftmul acquales erunt parabolas A B C ; £t fufpen^ 
datur figura G L H M e^ pundo D • 

AcGipiamriampun(5ia Q»& N a?qualiter diftantia il pun^ 
I, & E refpeaiue. Du^aifq. NQjsquidiflanterad EB,&R 
O S ad L M i Erit vt in praecedenclienimate N P (qualis ipfi 

RS. 

lam QN ad R S efl /ob jequalitatcm ) vt QN ad N P, fl* • 
uevt D/a ad ANrcciproce.. Aequiponderaiitergoredae QJLf».jtJ 
N , & R S • & fic femper • Ergo omnes ftmul linea? trianguli 
A JF C (nempeipfumtriangulum^ ajquipondcranc omnibus fi- 
jthul lineis figurap G L H M , ( oempe ipfi figur^ G L H M . ) 

Accipiatur AV teirtia pars totius AC. Manifefhimefl» . t 
quod (i ex V demittatur re<aa ^quidiftans ipfi A F . in ipfa erit 
centrumgrauitatis trianguli AFCi-^ritq.ipfa ad horizontem 
perpendicularis . Propterea erit triangulum A F C . appenfum 
cemralit^r ex pundo ,V . Eriiqj triangulura A F C ad fpatiuni 
G L H M • reciproce vt D A ad A V , nempe triplum . 

Cum autem fpatiurti G LH M a^quale fitparabolas A B Ci 
erit triangulum A F C triplum etiani parabola? A B C • ^^ 

UMmtSl^ ^od&c. 



p 



PropofithXXL 

Arabola fefquitenia efl triatiguli caackm ipfi bafuni 9t 
, caadeoiaijitudinemhabentis. 



Eflofemiparabola A6C> cuiu^idiameeer C£» ordinati 
AE, tangensverd CD, &compIeaturparaKelogrammum 
A £ C D . Maoifeflum eft quod omnes lincac trilinei mixti D 

L a ABC 




$4 De Dimenfioiie Parabolae 

A B Ct 'qwt atiidem diamctro paralle- 
Im fint» incer le funt in eadem ra&toneffl 
^pa fum omnes circuli com alieiiius» 
^uiaxeiiilubeac £yC»& verttcem C« 

M* ErgOMamingrauiatiscinniomlinea 
fumtnliaei DABC» erit in illa, quae 
diuiditlibram D C; quemadmodum 
diuidiceaiidem cemcum grauitatis co- 
ni>nempevcpars ad C teriiunatarre- 
hqixxiktrifh. Fiatergo CFcr^^ 
fius FD* &dudU FM paralldaad C 
£, eric cemram graukacts crdinct D A 
BC in reda FM . vbicuM]. fic. 

Item»omnesliness> cpM ia femiparabela ABCE dacaa- 
tur ad diametrum parallela? , inter fe funt in eadem ratioiie , ia 
quaiuntomnescirculiaiicuiusKemi^haerif^cQiiisaxi^ A£i 

*%u vertexveroA. Ergocentrumgrauitatisomniumiineanmiad 
libram A £ appenfaram, fiue if^us fem^arabola?» erickiilia, 
quae libram A E fic dimdit vt diuidit eaiidem ceflmmi gr»iica« 
tis bemifphaeri^;Nempe vt pars ad A temiinata» fit ad reGquam 
vc5^ad?.Fiatergo Alad lEvt^.adj^^&du^IH paral- 
lela ad C E » erii cenmun femiparabola? in reda I H , vbicm- 
quefit. Ducatw tandem GL, qua?bifiu*iamfecetlatefa AE, 
D C. & in G L erit centram grauicatis paraUelogranuni D E. 
quodfitOa Ponaturcentrumgrauttatfefemiparabolfeflrepun 
^uquoduis P.duiaaa. PO»producaturinN;&eritN.cemrfi 
grauitatis trilinei D ABC. Iam> femiparabola ad trilineum 
eftvt NO ad OP>lSueyt MLad LI;nempevt2.advnum; 
^qualium enim partium tota A E cft 8 , talium A M eft 2, M L 
eft i>LItfft vuat&teliqua IS ^.perconfttudionem^Ergo 
^ femiparabola ad parallelogrammum erit vt a. ad jt liiie vt ^«ad 
/ ; & femipara boia ad triangulum infcriptum vt 4. ad ^. Nemr 
pe feiquitenia ; Qupd&e* 



f I Jf I S. 



APPEN- 



Appendix^ 

De Dimenfione Cycloidis « 




IBET iic Sfftmdicis Ucs Mdderefahui^mm 
frotUmmJMM imMCMdi » . &J!mMeriJt, fr^ 
f$fiti0»€mifmtfftfits^ frimt isftmicM dijjUUi. 
mi . Tprfs hfK ^feftUisq.fUirihMs Mi bimc mn^ 
nis M^htmdncss ntSriffvulifrimMri^sfrm^ 
firMtnimttntdudtmtnSrMtto tttdfit db tllorH 
mdmihusobfdldcidmtxftritntif . Jifftmfismdmtqutdd librsm 
mdnufdSlamf^dt^sfigmrdrmmmdttridlibMs^^ioqHofdto^ td 
froforfioqMdvtritrifldtfiyftmtftrnmnorfmdtrifU dffdrmit. 
rndtfdifmm efi , ^iModfotims 4b fmfficiomem imcommenfmtdbi. 
Utdtis (vt tgo credo) qM/tm ob defftrdtionem demoomffrdtiomis , 
infiitMtd conttmfldtio dbilUs dmiffdfit . 
SMffofitMm tft hMiMfmo 
di.ConcifidturfMftrmd^ C^^ 

mtntt dhqMd rtSid lintd f>k**'**'^'***X 

a b^ . ctrcMlms a c , contim^ \j^ jv-i. 

gensrtBMn z.\>,inpun. ft ^& 

£io a . NotttMrq;fMn£fMm a , tdmqMdfixMminfmfhmrid circMU 
ac. TMm inttlligdtMrfMftrmdntnttrtefd a b.conMtrd circMlMm 
a c , motM circMUrifimMl drfrogrtj^MovtrfMSfdttts b .• itdvtfm 
bimdt dliqMofMifMncto rtStdlintdZ hftmfer cowtimgdt^^SMfdi^^ 
fixMmfMnciMm ittrMm ddcontd£tMm rtMtrtdfttr^fMtdin o . Cir^ 
tnm tH^tiodfMn^tMm a iixMm inftrifhfrid circMli rotdmtis a Ct 
dtiqMdm lintdm defcrihtt , fMrgtmttmfrimb dfmbieSd Untd a b, 
deindecmlmindnttm vtrfns d ,• fofhtmofr&ndm , defcendettm^ 
qmtvtrfMsfMnCtMm b. 

VocdtdtB ifrddecefforibMsnoBris . PrdcifMe iGdliteo idnp 
fttfrd 4s. dnmmmh hmitifmodi Umtd a d b» Cyclois^ rtttdvtr^ a b. 
bdfis cycloidis ,• Jt drcmlMs a c , genitor cyctoidis . 
Frofrittds , dr ndtmrd cycto^dis td tfi , vt bdfis iffins a b.f - 

qMdlis 



S6 Appcndix 

qffdlis Jit feripkdrU circuU gemtoris ac. JSit^dqHUfm n§m 
ddeh obJcHrum eH < Ndm taatferifh^ria ^cfe iffam m cmmtcr^ 
fone commenfnrMitfufer mdnente reifd a b . 

£uJLritttr punc qudmfrofortioftfm kdbedtffdtittmcycUidd* 
/^ adb ddcirculumfuumgenitorem ac f Ojl€hdemufque,I>e0 
dldnte 9 triflum tjfe • DemonHrdtiones rres erant , imerfefetU* 
.ttu diuerfde . frimd , & tertidfer. noudm Indiuijibilium Ge^me- 
Jridm nobis dmicifftmdmfrocedent :fecundd verbfer dttftUem 
fojitioncm , mare veterum rectfto ; n>tvtTifquefdutoribusJatif 
fdt . Cfterum , hoc moneo ;frincifidfere omnid » quibtis ddiqmid 
fer IndiuisibiUumGeometridmdetmniirdtttr y ddfolitdm dttti^ 
quorum dcmonlirdtionem indire£fdm reducifojfe : quodi npbis 
fd£Ium eliy vt in multis dlqs^ itd etidm infrim^^ c^ tertiofeqmcn 
tium Theoremdtum^fedne le£Ioris fdtientid nimium ddhttc dbu 
teremur flurd omittendd cenfuimus ytrefq^tdntHmdem^ttfird^ 
tiones exib emus ^ 



THEOREMA I. 

Omne fpatium quod fub linca Cydoidc , & rc<5ta cius baif 
continctur/triplumeftcirculifuigcnitoriii fiue fcfquialtcruni 
trianguli eandem baiim j & eandcm alticudincm habcntis • 

Ejfo Cyclois lined a b c de^ 
fcriftdJfuniio c circuli cd 
c { dum iffe circumuertitur 
fufer mdnense bdsi a f . (con- 
siderdmus dutem femicychi-^ 
demydrfemicircuUtm tdntum 
dd euitdnddm figurd confu" 
sionem.) Dicciffdtium a b c 
itriflum effeftmicircuU cA 
cf; siuefejquidlterumtridnguli acf, 

Accifidnturduo funcld h , ^ i indidmetro c f . dqu^remo- 
tddcentro g • Du^ifq. h b, il c m uquidiJidnterifsiU.trdn 
fedntferfunifdby&\femicircnli obp, min, fqudes ifsic 
dit&^otttingentesbdsiminfun&is^Xki. 

Ndn^* 




De-Cycloide, 87 

lidmfeHum ejireiids hd,ie,xb,ql ^qndles ejfe ,per 1 4 
Tertijy^qMdlefq. eruntarcsis o b , I n • Item cum^qMalessint ch 
i f , fqudles erunt c r , u 3 obfardleUs . 

Totaperi^frid m i n , ^^ cycloidem, dqudlis ejhrecif x f. ite^ 
^ue drcus 1 n recfd an ob edndem cdufnm , cum drcus 1 n feip^^ 
fumfuper reifd a n commenfurduerit i ergo reliquus drcus 1 m , 
reliqud rectf n f fqudlis erit . Eddem rdtione drcus b p . recif 
ap, drdrcus bo reiff pf, dqudliserity 

Jdm reifd a n dqudlis eB drcui 1 n jyZs^r drcui b o , jJV^ r^(!?f 
p f . J?r^^ obpardllelds , fqudles erunt dt^ ic. Verum quid d^ 
^udles erdnt etidm c r , a u . rdiquf u t , f r fqudles erunt . Pro-- 
ptered in tridngulis dequidngulis u t q , r f x , dequdlid erunt Id^ 
terd homologd u q , x r . Pdtet itdque quoddude reifd 1 u , b r ^- 
mulfumftdedequdles erunt dudbus reilis 1 q^ b x^, nompe iffis 
c i , d h , c^ hocfemfer verum eritvbicunqfumdntmr duo pun^ 
ifd h yt^ ijdumodo dequdliter d centrofntremotd.^ Mrgo om^ . 
nes iinedefigurde a 1 b c a dequdlesfunl omnibuslineisfemicir . 
€uli c d e f i c^ idelfigurd bilinedris a 1 b c a dequdUs eritfemi* . 
iirculo c d e f . 

Sed tridngulum a c f duplum efifemicirculi c<i e f . (nttm tri 
dngulum ncireciprocumefi tridngulo Propofpr.] Arch.de di^- 
menfcirc. cumldtus ^affemiperiphderiae^ latusvno £c did* 
metrofitaqudleyvndefequiturtridngulum acfaequaie ejfein^ 
tegrocircu/o cuiusdiameterfit c i.) Ergocomponendo^ totum 
€yci4idalefp4tiumfefquialterum erit trianguli infcripti acf j 
TripUtmvtro femictrculi c d e f * Mnpdtrat* 

Lemma I. 

Si fuper Iraenbus oppofitis alicuius re(5tanguli A F » duo it- 
micirciilidefcripti finr, E I F , A G D erit figura fub periph«- 
rij^ & fub reliquis lateribiis compr^henfa ^qualis predi<ao re» 
ftuiguIo« 

Vocetur autem talis figura Arcuatum j fOmfifuerit integra , 
ofuametiamipfiuspartis y fuandoftifAfueritalinea ipsi fd 

paralltU^ « 

Htmon'^ 



8S Appendix 

Dem$nHfdiw ; ijfHMUm cum sint MtquM^ 
UsftmicircMmfti cammnniftgmtM b g c, 
ddditifqnt commtinibn^ trilintis e b a , C f d • 
cUrum erit fTopositHm . 

^Ando vtro dttur cdfus quod ftgmemttmo^ 
mtdlum sitytunc brtuiorfdcUiorq. dtmonfird^ 
tU trit . Fdcilt ttidmftr tdndtm frofidfkt" 
rtfim ojfenditur drcudtum fectum d lintd iffi 
f d fdtdUtU fqudlt tfe rectdnguU dqutdba^ 
^fufcr tddtm bdJiconHituto . 





^ _ 

BfiQ lintd cy^ 
ckiddlis ft b C 
dtfcriftd )tfuso 
do c femicircso 
licdtdumcon 
utftitur fufot 
mdnentt ac« 
ComfUdtur rt^ 
Sfdugulu^ f c Cy 

fdtq. circd dimttrum a iftnucircuUs a g f . I>ico tfddidtm 
a b c fecdte hifdridm drcudtum a g f c d e . 

Si tnim itd njon tfly trit vtiq. dlttrum tx duobtts triHsoeit f ga 
hc'i Sihcdc,mdgisqudmdimidiumtiu/dtmdrcudti. Bfi^ & 
fondtur dlttrum txifsis (quodeumi.sit) fmtd zhcdc mndus 
qudm dimidium drcuati . Sitq. txce/pis , quo trilintum fuftrdt 
femiffcmdrcudtiydtqudlisffdtiocuiddm K. 

Stccturbtfioridm ^cin\\i& ittritm h c i* i .• &si€fidtftm 
firddntc rtStdt^tdtm dliquod i e c minus rtftridtmrffdtU K. 
Tmnc diuiddturimttgrd a e infurticulds dqudtes ifsi i c , ti^fet 
fun^ddiuisionum l,h,i, trdnfednt femicircuU dequMtrifsi 
cdc femicircuU ytdt^tmttsbdsiminfunais Uhyi.ftcdmte/q. 

cjcUidtmim o,h ^m^ftr qudt fmmctddg^mtmrrtctm^o^ifh^ 

q m d dtquidifidntts bdsi a c • 

tEri 




Mr^ hS^Me MtCMMimm o h fqnAle iffi^l: MriMtam ver} h i 
ffMMe drcMdU p h : ^ arcu^mm m e acimdle drcudto q i « i»r#« 
ftd^tA 'mnmiffdfigwdinfcTilftd i» trilinep a b c d e cenfidns ex 
arcMMtis , fqiudis eritfignrdHdemtrilifieo circnmfcriptfj exce^ 
fto tdmen arcudto 1 m rc d e • ^odfifigura circnmfcriftd dd* 
d^fkmm nrcuntnm i m r c d tfuftrsAit circumfiriftdfigurd ifm 
fam infcriftdm cxctffufrddiSH arcudtifiue recidnguio r e> nem 
fcminofi exceffu quMnfttffdtium K . Proftereu infcriftuin tri^ 
UnepfignrAudhuc eritfln/qu^dimidium drcudti a g f c d e.^ 
ideo mdior qudm trilinetan f g a b c • Sed eddem aqudus eH dlte* ffivtJU* 
rifigttrf ex drcudtis comfofitf & in trilineo f g a o c defcriftd : tur mfio 
ergb hfC infcriftdfigurdmdior effetfuo trilineo fgabc. fdrt 
fuototo.quodeffenonfotefi. - . 

^ ^noi infcriftdt figturdfint fqUdlesfdtft.Nam drcus o Vfqud^ 
UseH reShe { a , hoc 'efi re&de i e , hfc efidtcui r m ( ^^ cyctoi^ 
dem.) Ergo drcudtttm oYi dequdUeritdrcudtotxxl. &ficdi 
fi/Ogecks. ,', . ■ .'.\-v^v;..V ~ r.r.y\\ -, • ■ 
. SJvt^ofufi^Uit^eimtJ^t^iUn^ mdius quam dimi^ 

dinm drcudti a g fc 4c f- ^fttftrnaiofigurdey (jr demonHrdtio 
ffnit^seddemerit. ErfffiCopclude^t(4cycUidemUnedm abc 
4^mmfecdrf, drfu^tunt^agf^: dc . Sfpd er^ frofofitom . \, 






» \ 



r^^J^,ojfjBM4 //. 



. Spatiumcycloidaletriplufneft 

^jrculifui genit<>ri&« 

J^c/cUfsahcd^criffddfii 

ctoxctrcp^cfd.dico,^dtium a . 

^d.mf,lii:tfiefemicircnU c,^A^ 
. Qott^lM4m<reftdttgtdunt jid 
cki fdctoqfufer zt femicirctt^ 

te sLgeyducdturzc. 

Triattgulum a d c duflum efffefni^ircuU c f d {ndm hdfis a d 
dequdlis efiferifhderide c f d oi cycUidem , dUitudo wro d c 
dqudUs didmetro) ide}tre£fdngulum ed quddruflum eriteiuf 




f> '"• •> 



M 



dcm 



♦i* 



dtfh ferifiviti:«H cftl; fir^ »rckJititm ig«tf^ pHttHi^lMm 
nritfitf/SempynicirnXi :^pmfi^fithiHm zhtf^^ffritm. 

ihkfrsceihHs) duplnik rfrifp^icirej^ ^ t^ ^jjuy m-Mtltj^nttw» 
a b t d i(rif>iifm mt -AttfSemJmh^eiiH ^/^ . 



T»SVli£*Jif'J ttt. 



' ^Ottttit %irium tyclcA 
Ssittri^ititi rtl tirCuK 



> • 



^ IWl T 




,/r^ i w^^S^C^TJ 








rV. Nw-lX^ 1 



t 

H 



A 4^ 



<4 



itii gettitoriS. 



tffffcythiidjB }ine}t 
a b c defcripia a Puft^o 
t JiMicirciiii c € d . 2)» 
WffTUfihn abtd «Tjp^- 

Compleatttr reBdttguUtm a f cd \f4^tq.femicirc«h «^^ 

Vl^ ,^^4fl»^jt»i!^ 'h4 ,1% dfefiiHimmtetlid ^ d . fim^^y^Mm 
ifki»ti*ik tjtiOmfitis-vifSefis ^ «i^* ^ . 'Ji(|^i»^ idietti Mmjm V» 

/^ m?4 g o , eqkJtUs\fh^f tHi>(>Mf)» iqualesjittt g r , o 
u , ^ commuttis r o ^y^ivr ai^UMlis efi reiff a n , tiemfe arcui o o 
(^*^ cycleidemjveiarcui p b ,^t^Ye&f "^it, w/t h , «H^l^jtf. 

Ebdcmfrorfusmtdo , tjuo detho»Hrauim0Sfe&am ^o 4yw 
i«7» ^ r^<?iff bf, dimouRr/iittm^timtH^s^fiitgt^e ilui^trtri- 
linei fg a b c ^e^alts imuibMs itttiiPtHUitH a' b c e d . yMjMr 
teadictatrMtte4--ittUrftitequMiai'eiii^SBfgtV9 ^f Ki(»*d4it' 
ti Thetremate 7l*umtBriiiitur *ey&iditatt jfmim " W^ Jmitffe 
JemicirculiCfti . ^S^oderatHrc, x 



y*\fr f 4. 



» ' 



^CA^O' 



I 

t 

i 

» 

v 




S CHOLIVM* 

Cyctatdibii$ allaifiifii ^ecie* 

ti$H€mhd9m 4tmm^4i^f»$frsk^, $^i^um p/ti , d? 
0memffmt0dicebkmjUitet'^ I^9fmm$mmt^pH€9^4^4uUmf 
0m4mK£Akdmcjigm^m$f^tmMmy ftfmm Cjtelfi^im Mtimmmm ^ 
feUmm mm ^fjff mmmnmhMf ^ f$(mmie ^ mi dm imfimifmmm 
ff^eies hmmjfmm^ Jj ^mm t mm mk iffim iim. 4i^i^fi4erjtfm fitimtmid 
Cjtei^dtim mum m , C m ^ if ^m mtmmmiimjigm^l^i ^^ 
mmf /jJemfatifhifHmmfi^mli ^ii mi^mmh^itmifir^HmmemiB 
fuLeMMm^mmmirfi^ qf$4m eoU^mmm ift^ 

fims circuli ac im imfimitmm t^xtemfi . MAmifefimn efi ptedfiifcm^ 
Xitrr»/rpiMfiU»/ttNM^ h 4mf0iMiimHem.Pmm 

Mmfuem^fmmtimmMfmifh^^ 

defctihcmihmmtiUmtfmmmiffdfitifmmhit »41) ^^^44m etidm 
ifM^ uiert^ikqfm^vidmwyfit^t^ ^vem dd edfdm 
fettcs concdmmm hdkejmeat;. t^ditfmfi^mP^f^fi^ff^^ ^e^ 
mmmMircmiirhmftif ao ^idd^mtih^ . PfmM "^e^ fm^ex^ 
mmfmhfJtmrimm^ ac ermmttCychtide^defv^ihe^fiyjffdfrimf^N^ 

mk^emji y.&nfqmi9tmfm^ 

. eitcjdmmetutfe.tm^^ejtcliiidisfe^ 

fmmmes eifmyjemimtfwif^hmettim fem^mrie4 fi^ fmfkdctime «t p ^ 

immfemtqL^fmnMtemm^iyitl/^emiffmm dff^mfmtf .. 

Mhmc cmmmemimmtmmej^fm^dMit4iMmiift^ 
hmmtHimlhs umumcyeleidtthJifim hdmfttfHt gemitrifi ftfiflfidtfid 
T mtjmtem : sttktes vere mimrem^mijriei ferifhderid bdfifm hdr^ 
bent • 

Rmin , ^mmmvmdjfmdefi cyeieiddUefigmt hdheJ ddfmttm tri- 
dm gmim m ^ meLddciremlmtmfstmmftefr.itim ^mfftrem , //mftrefi 
mdioris infqudUtdtis^ wbridimtimimfimHmm . Si tdmt^n vtrmm' 

i M 2 que 



1" 



j;l Appendix 

tUkr&X V it V.^ v 7 Jl 1 vl/ C^ 

Oiline fpatium fub qualibet cycloide linea, & reda eius bafi 
contentaar,4rd^aflg^to alti. 

tudine conftimftim , eft vt peripna?ria circuli prnprij gemtoris 
vna cum duplo bafis cycldidis , ad^uplum bafis cycloidis • Ad 
circulumveropropriumgenitotem vnumquodquc cycloidalc 
i^ariufn^eft vt duplum ba mcydoidis vm cumfer^rfiaoia ]^np 
fbrisciitaliadeiufiicmcJrn^fiperitfksRi^^ * ^ \ 

icrirmf^mditMis^ cycUidjtlt ^tmt^ituifttin:^ ijMmistd.tnMm^ 
^HlMtm yjSmt tircttltitttfkMmtfit im tj^tttrttcitttey <^ im dm$jth^. 
' Cuiukiiiiq. cycioidaHsfpaQj 2^ quodiiba %ftciiuii q^doiddk- 
tc <^€titmtfitt$tt db eddttmfirm ttrimtychidctT t ttm t d me dtt t ) ratio 
iC^iDponitur cx ratione altimdiiiis^advakkud;)^ & ex radone du- 
piibafi^cymper^phamageiiitn^ bafis cunperi* 

l*ifj'iagcmtrice. ^ .' >/ 

' Ttitigemtem Md)qttedUbef itmfermtmtmfttmSmmt dmriftjfe certi 
V/f ;fecMli4ri primmm rdtitttefrtCjcUidefrmttMriSideittde wti^ 
^erfali etidimfre mmmibtts tiiifs . Tangens ad danim quodlibcc 
pundum primaria? cycloidis ducitur ex pundo^fublimiori gcoi- 
-toris circuli per ipfum datiAn pun^um «anfeumis. 
' Titngens ad datuni punAum cuiufeunqu^ eycloidis ducitnr 
tHO: modo ..Ttanieat per datum pttn^m<:ycloidiscirciiliisip 
fius geniror , quem in dato eodem pmSto cotitiogat redacoo* 
^ti€f)iens vel cum bafi^ycloidis > vel cum alia ipfi acquidiftante. 
f iitq; vt radiuscirculipopri) ad radiumcircHK>pmnari| , itatan 
gens pro^didhi inter<}atum pun^tum, 8c bafkn > vchB^diftaiiie 
Sifercepta^ad aliam qnandam lineam apte Xumendam a tcmii* 
ho tangcntis in ipfa vcl baii, vel squidifiante • Tum ab^trcmi 
tdtthuiusatrumpta? tangens ad imperatum punftum. cycloidis 
emittatur. , 

' Wtmtttdlit itimiit Them^Mtdtd frp MesMtUcis^ cemee^tMeri' 
hts €x isdc fgurAderiuMriftffkttty tti^^eufidemdttuiimmttttmdmm. 
i(JitmcJmulcHtnm9$UftimitdimtmfiMty 

F I N I S. 



.:i'\-:*s\v^i1 



I #< ' 



9S 



« * . » • I 



I. 



D E 



u. 





♦ t 



HYPERBOLICO 

Probtem» dtmim • 



- ■ ■ ■ • m 

Troimmfn MiLeBorem . 



^^•4 




GdREDiOR iamopdsqaodipliisGcomcr 
triae candrdatis non foJbini difiicile videatur vc 
nixn ctiam impoflibile .' Hadenus cnim in Ma- 
ihematicis Scholjs repcrtap funt dimenSones 
figuVhmii ab omni paite fihem habentiumr 
quandoc^uicleni intcr omnia folidla, quae ab an^ 
tiquis , & mod^irrjis Ai3ift6rife!us multipfici conawad mchfuram 
redada funt;.nullum adfiuc ,quod ego fciafti, vllanrdimtnffo^ 
ncm habuit extcnfione infinitam. Iii^ ftatimate;pfopx>na^ 
tur fiue folidum aliquod; fiuc figuta pl^na , cuins anqua cxtcn- 
fio in infinitam diftantiam proceddt, vniliquifque cbgitabh ht^ 
iufmodi figuram infinitg magnitudinis cfle dcbere^^ AtrameiJ 
folidum habct Gcemctria , longimdinc quidem infihitnm , fed 
tanta pteditum fubtilitaie , vtficer in ihfinitumproducarar, eiiJ 
gui tamcn cylindri molcm non exccdat . Talc erit folidum illud 
abhypcrbola gcnitum,quod huius libelK cohtcmpratione pro- 
fequcmur ; intafliim hucufique ab alij[s,.& multip&:i,curidlaquc 
Thcorcmatum varictatc fecundiflimum; c6* vfq. vt, nifi me ftl* 

lat 



M JPromium. 

txi affeAos» vniuerfa Geometria incer hadenus confideratas fi« 
guras MiUam habeat curiofiiMs almdanciorem • 
Quo ad methodum demottftrancfl ; vnicum quidem , & pr^- 
^ii^j^X^ 6c;^|;ia(^ii^- 

iilij^> &|norc^^t£ruii «^^uam^aA ^v|v^a ^cf a^iuiy fd- 
m5 mudtftum (k pcMftdlllJfiDiiium ^oitl^ttiato^^EjOf fan^ verus 
eft demj^i^jU^difiodus .ii^ei)4fic(^ (9^^^ ^if^S^^ &ipfi 
natura^ gMVnMf&s. Miltrelflfte «t«ieff^G^4malriaiUiu* «u Indi- 
uifibilium dodtrinam, fiue non nouerit , fiue non admiferit ^ cir- 
cadimenfioae^A^li4Qn|tH a^l^a^^i^icates inuenir, vt 
ipsa penuria infclik acf (Etaterii nouram peruenerit . Antiquo- 
rum enim T) leoremaca circd dodijQam folidorum , quota par$ 
funtcontemplationum,i<]pts0ffi^^ aeuo Caualerius 

f omflis alijs^ inftituitjciici tof aarici5folidorum,fpecie diiferc- 
tium, multitudin^ abun<^ptiiu|i ? ^ethodus r^^ftra^quam vfur- 
paturi fumus in pr^fato Thearehlate,pf ocJedet per Indiuifibilia 
curua , fine aiiorum exemplo, non tamen fine prxmifia Geome 
tt^seapprQb^^tipaei, Confidcrabic^i^ e9i0i9mn^.c)dliQddctf 
(ipcrnciescirca communem axem in hoftro fplidtxk^r^bi- 
Ijei^* Cuius reicum miUum Caualerius ipfe tradidaric in fui 
Gco9ictriaelcmentum»exiAiiiu.iumus noftr^m tt^aeo^t ra- 
liQqqpi exemjpjis aliquot cfle cqrroborand^m » Qi^mquam 
b^c^pudtpcuiperfiuumfitscuaxiam, toti^i;!! huius itbciripo^ 
|f cflutnratqm babq^oi » eo quod i^fuiTi a^miferit ,^probauerit- 
quK do(^(j[iau)s.^& eruditiflimus vir MofhaclHd^hnu^ ; cui,vt 
ip. pluriiius alijs fcienti js , mihds^, ita & k Mathematicis di- 
fciplinis ncQiinem quis iure antt'pofi|ie«it . Pra^mfttemus ita- 
que 4nce ipfuqa. pjpus , iv\h Exem^Iorum nomine , quaiciain 
,Geomctrr^ pi;opofitiones iampridem ivotas,fed a nobis pcr In- 
diuifibiiiia curua demonflratas: Sic ehim magis manifcft um fiet 
hunc modum demonflrandi non efle negligendum , pr^fenim 
cum in rebus difficillimis maximum ipfius momenmm reperia- 
tur • Indiuifibilia vero curua qux ad huiufmodi demonftracio* 
nes idonea iujQC , in planis quidcm figuris iolje cirCulorum peri^ 
pha?rig fe fe oiferunci in fohdis autem , lup^rficies fphxric2e,CY* 

lin- 



lindricae^cohicaequc. (^gmiAqiii amMm M mwit rnN H rw t- 
biles funt, tamquatn ipfa^ fignras (Hffift^ aJ a q u i ^ v &«•» 
dique aequalis , y nifbra^^uc<^ itatdittoij ^^buti^ . fr^ 
mittimus igimr lante opeits ^ggn^otkmiftoaafikitacipot 
Thec^tamtwa^ometrioonn ^BwnifUjt* 



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t$tr dequ/dis ftTifh*ri/t b d. T» 

eoHiMgdtitr ac. Dic^ehreM' 

Itfmhd. tridfritiU abc «^ 

fquAUm . 

S$tm4»iiri»femtdiMitan mib , f rti /i >fa >j»»/> ^ W iiiii f ^rtfygiy 
4g4»tiirif^^^d iotekkmMtmttitmMkt^i.^imBiK^i^ 
tMttUdd hc. fykitdfmtfmfbdBrJM hd^^Mifftnif^miiitm tifti 

ptfijmduimetit b.AV^ac. nfjimm^ttvrHimjimp-Jfmni^ 
tto/iji/fptih^jditekli dimettftone)fime mh c «irij? ||f |i ■— m i 
/Wtf > erit ferifhmd h^^ ttdkm^iMafmtfibii^w^ dd 
rtMk-iak.- Mjgkfmriphfrim^iit^ititMm U viwf mf^mlitsvd', htg 

fmtiifiimft f t« mei ii tta ' »<^r t ^ mit ifi»mftm m ■ai A i i. i 

« . • • ••«■*••* * ■ » •' ^ *' ■» •• • • 

iwvtfatffhdtrd b f , &conusrilfus c a d ; hiccffhfram b f, 
tf ^;!r^ ctlA ejfe dtqudiUm . Sumdtur enim imh qu§duis pumfu i> 
l^ftr iffttm i tr4tifedtfuferfcies/fh(ricd\ b > wri ettttrum a; 





l^tAffhpm^^ ddffk^rmfm 

iaJ^^ii&vi,f0Mdt^fm ha^ ^if$^4iii[ 

9, i ifi^€ v$ qtiddrdium \c. dd^iudh il i * 
new$fevtcirctilMscA^ ddcirculMmlm. 
Stddmtecidentyfqdd^sfMr^^efeH^ ' ^ 
c$n/eqneMtes : nemfeffhfticdfuferfici 

es ihyfftfMiseriteircnlfrlm* ^imff^t^^^Hl^Mmf^^fit 
funStum i . Profterpgcm^esffhdm^f^q^ 
iffd sfhderd b i) de{u4^s &un\ on mij^ t^irfj^sfi^f^fuimfiis^ 
fiueconoc\^..:'^!pderdt(irci ^ 




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\^ f^h d it^ .^<cuku dimmeia <ai>» t 
tdnremff.\iAfit^:def0diisfhnidiM} \^\ 
poesrOff^mdirdiEt conbtfthdVL^ ^ cmuer 
fdtttrtiidmgtd. zAh eifcd dxetn bd. 
itdmtfidtconusreSusxAc^^ ^ , ^ \ \ 
'^ ^Jiicoffiid€rdmzhdefttdiam\i^^O'*\ >\m. \>\\i.i \. .; 

M ailc. Stmmdistr£nimi9ididmem^'$,\Hit^u^fmt$&mm u 
fetfMddtrdnfemMrcuits ih^Mimkenudre&me^ff^^ 
fefficiessylindricM 1 i mn ^ ckrcd dxem.d b imcome\ 
^fextu ' iitmttcmmzhdu^fitiffius hA^erit zildtq^ ill «35* 

%.t-ie rcifdnguU 1 i b , cf dequdte reSidngulo 1 i m . Profteu^ erit cit 

Jblsdu culus f h defudhsfuferficiei cylindricde 1 i m n . Et t^cfiem- 

ff^^* fer , vticunfuefitfuniiiak iv ; Ergo-omies circulifimulyfime if^ 

fdffhfrd , ffUdles erunt omnibusfuferficiebus cyUndricis fimul 

fumf$is^}iemfeiffic4n$^6x:\ S^$djt$n€$fddt)ctmt i%.iiXf.f. 

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UHirus. h l,,&,Hntis ^ c C- _^.S*tmd^4cifr4e4nrtai^^^Cy^iud' 
misfM$$&im h ; fcr qiif^int^l^gMm a^tf^ f^fcftfttf^l, if^^ 
contmcpmfr^thenfms i ^ infiifer fMerpies eylim^icdt enint 
/mpfnhdxi/veri ctt. Mriteriifiii^fAes^tnmd hi %^ 
ddtireMlmn h Xfndm Ufimy v* re^A hUdd fndrtdmpdrtem Sfbm, 
didmefrihl £tAoevermneritfem/>erytiiem»fifafnn£hnmh» 
Erg9emnes/mnlfnferjft!^esxjksidiaem{nemfef4lidnmyam$dex 
ejUmdrerelimfmitmr t dcmftecdd» aicfjmd tm m e s fama tirem-' 

^' (%<^» ^^mtm^c £j «»«f,» ^tfmtemmtis^fimmlreaf 
tria^giiUj^hz , ddfmdrtdmfdrt^m^en^tiiimrJ&^rMmi^p^ 
Sicf; nen^ iii. vdtioi^dmfU^ ^ed eeneerddt tmm Thtmram^ 
u xM.Xii:EmeUdis, .. ^ / ^ ♦ «^c ^ 






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(irffodu£ia hc mit^iidvtcircMltts^ 
^ki$isdidm€ttr • c e ^Jttfijttalis etsrHttftf^ 
jfjpr^iH^ni ^hc^VQncifiAStttrcirfjtdit^, 
\metrtt9tiC^€ifC^tts€r€aMSdtdfl4ttMt9t^^ LS^ 
}) i}&f^^irSMh ,c e, itgs^UigMtttr mU 
tacint^ ,c d e ^yhdb^ns v€rtictmitt d, 

J[)iC0 C^MMttt a b C , tOH€ C d C * ^ fqMM^ 

Itttt^ 

Snmdtnr inre^i d c ywijWw pnnifnm h , fefmmed dmifi^ 
h AfdrdM^ *rf b c , inteUigdtmrf^ h %firfeffieUiemid $. 




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>8 

g h ; €irciq\ iffdm h n chcMtMsffdrMUelMs circuU c e • 
nicAfuftfpctes abc, adcircmism/mdiMfis ^.c^eHvtrcelM bc 
MdcAyfiMe^vt gh ^^hd; nemfe^ytceMicMfmftrfi^icsm.^^ 
dihrcMlMm hiiu. CjfcMits MMter^ ic-^ Mdci»ci!htmt '^i^eBvt 
qM^MflMm fMdjsbrhti d c , Md^ifiiStt^.^^c t ijkfc n^ipUMimflmm 
qMddrdii d h , MdqMddrdtixii thec tff vC^CilN^tdMS m h^ JtdcircM 
iMtH H n • Brgb ex dqM$ , erit c^micdfMferfiHes a b c , dd,circM* 
Uim c e , "vtconicd m g h , ddcircttlMmh n .^kMcfemtper vt- 
IrUim erijt yn;hicMnf;fkcritfkm&mm h • £r|l mmtmesjmemlcnicf 
fnferfcitsimefnftcenMs^hc) dqmdlestrMht^mmibmsfimemlcir 

\ms ynemife^0i^$x d e^- ^odrrdtdrc. 

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ES/i> tirhdk/^ciUMs iidmeti^ a b lifimdtmrq^e t^gems bc 
didmei. fqditliSy driMnSf^z c, cenMtrtdtmrfigMrMcircM dxi 
a U» '^^ vtfidtffhxrd a c b f , c^ c^nMsre^Ms c a n • 

Z)/r^ r^/i/ir«r c a n , iffiMsffhdSM dMf^ 
iMmeJfe. ^ 

AccifidtMr in didmetrd a b qModSBet 

fMnStMm d , /^rr qMod dgdtMr fldnMm c f . 

dddxem a b ereiittm ; fModqMidem fld^ 

nmm dMOs ctrcnlos e^ iet^ alterMm c f « 

j^yj' ^iT, ^lterum vero hiinc cno ; Conci" 

5. p; Je fidtsirfHfcrbdfi h i cflindrMsreStMs h t m i • Tdm t fMferfcies 

j9U:ffk: cjlindri h 1 m i ,.44t clrcMtum cfy cB i/t reStdngMtkm Uydd 

MMddrdtMm e d ; ^^^«r^r ^ij^/^ . £t hocfemfer ; vbicMnq. fitfnn 

{fMm d : froftereayVt vna ddvfiiy ita omnes ddemnes.ErMnt cr- 

go omnes fMferficies cylindricM ^ nemfe conms can, dd cmnts 

circulos , nemfeddffhpram a e b f , in ratiome dnfla . ^Moddrc. 

• , Cmc^tdafftm hec ^heoremati! 'Pr^fojjlf.^. ?.^/}i' ^^ ^tSt-^tf,^' V^*^-' • 

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y^ -Tw cmcfiksy C9iun dumtttf ab, *^g^q\ icfdndittr 
r^ diAmttf$ d^ti4liftt(piifmOd^} Cijt coHuertdiMrfgtifd ch^ 
€i MX0m^i^tii4ifiAk{€m^t4»gmtiic ^itdyt^citCiilQ de/ku- 
haturffhdrd^imAHgnU vfrtiakCtfpU^. ' : 

Wifria» qaodddm cylindrictint e^^cdMdtnm 
demfto c$i9$ c e d • Dicefphfrnmfradi^ 
iiofelidocxcdiidtetffcaqiiMlcm^ 

SdtmAturiti didtnetrtLSL b qntdtfisptitt 
Umm h \ferq^edinteUigM$irfiiftrfcits 
ffh^kd h i , ^r/>r/ fuferficiei ffhffP' 
€^ ^ctncentricd i (^ infttfer fnftrf^ T! 

rr» cyU^icd , f /ir^ defcribitUT d rtCld^ C K^. 

hYtdfigeiiti^c fdTiittddJc^rcd dxe)^t£ . 
rf<r#/*/^. /4;w> ^2.^d^tejivt Ilji^be^i^ 

^ h^ ^y4rii//y w/ ,• t^ ideofuferfiCits ^udsyUmdri^ 1 b io/to*^ 
listritfiiftrficitiffhdrjCdJ^i. ^(WM(cr,^<f^^^ 
fHniiumh.Profteredomnes omnibus^ntmft $mncs fuftrficiii 
fff^mcffimul ^fruejfhcrd -^ b ^(^UdleJ^ru^. ^mt^M: fttffitfi^ 
€iebuscyUndrictsfimuU hocefifoUdo excdUdtOQ jtb^ ^dmfi» 




\ 



Xttee»rdats$miH Thtw. fnffik }tMSfiin»rv ^limdn, 

\ 

Sl^mma. 
Vfcrpcies cMittf<iin'(iite fylfHdri 7<', , 
,ffi,i,h(i/tteJU^ojftmferfiKc}^. & 

,bfts) adjujterpiem CKrMtit c lutt/ci^ntiuf 1 71 
[egmeiififfharifi c d e , <•/ w rtUahfft^ J l 
lum fer Axtm cyliadri t att reiiaitgitUim A 
i^iifMbeatenfegmetttiy&diametroffhfra. 

Uam yfttferfieieseykndrifa a b , ttdci9^itiim<mti^f€0odu$r 
meterfitlinedexfolo ^c,efivi.r€aa»^M}ttm.^\it ^ qMHdr^ 




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inm dcs Bfgkyf^mfHiffttPiitiMiimfqMMti^t, 9nt cyB^ 
dncdfitferjuiet , 4deMd]fhWc¥^mei$ti c d c fkferf^it^ 
^ugditldmi^^fdrea^fidmmi^X. ^adt&c. 

Xl dKufit reBd c d ; St^eHtttr^i f^tu a b '4d Mkem -ereO» . 
Dit^mA^m 3ih//ijr4^i4it^*fi ep hmfmens^f^kfriti 

t ijlccspunm cKeoMdlis tpfi T^ (* tx 

cp.i ^nhuUtgiUur c\ltmYus\ 

qm€fqsuUscjlisidf§2i\iXm^ 'A 
aifUtmetistm dtsitfttti coUits ^"^ V' 

\%TK&fiafift0t^t$ ^J^<jc g|)wr;^; ^; ^ 

misftttsae^o jRdtUdqitntey^p^C,^^'^ 

^fe^*ifr^;^c}^iid^ekMtfl 
it^fekde^eylhtikicft lc d nt rjr^ • * ^" ' yjj^ 
r -^^iiHs^Utiett^cottfX^^rxxl/it .^" 

ftit4^Jf^fic^fti^eTficm 

^U^fime^ppt^^^^'^^^ ' .. .-/^•^^-"^ • 
T' iMtttmm c^y ^tet^terectane g a, ri? -r/ e% /aT gcf w»r 
.; feyVtabld^^Xk^ itSd %^^idtowetitfiek c^vti^dd o^p ^ «*^' v^ ^^ 
y. M c g , iie^ g O ,\A*^ ^/ c 1 , Wo i ., Seddsttecedettstsfutti dqttd^ 

tesyerjne^reiffo^yoifqttdteserttnt^ Froffere)^rectA9tgwl4r 
^ oPr<ioi dqu^erHttfiS^meffbci^r^^H^ 

''^ ficihfiir(9tSt^^lfdMTfector tc^^ydjttafeseritsit ^mmhttsfd^ 
feificiehtsc^nirifirfimMifi/i^tir^ her eflfo&io exckitdto 1 c 
, dm» Cttifidddatmconttst4mabldttts\^m^pdteBitfrofofi^ 

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tt .,.,.\i ,-..,. r.1,,',»» v\««iBiiiin»,tii-..\_\ 
.■.■o«.iiV,«i. .i>.vt. >.,,.«,..(.. -. 

■Tl&MUlte^eMitTmemnm^gilcly. . C~ 

agb. Sritigittnrjretis abA^ch» ttt 

femiJiiimeteti^jdtiii'fetltii^lttiiitdci. 

c fh , w/ a g b ,4ddHiHlHin c f d . ErgtfHlenrcHHt a b , ,<,^c d, 

liffitit/miiUere eemff/ftiimetcriaieSiiimfiaii^i^ftnf^i^V^ 

^Qf^di>^Himim'X'^iy>iMc£i\^ ^ ■•. .' ! .IZ 

r-L ..J -^' ^fitnsfhimi.V^IIL ,.Mii V .*««?, 

■j .1 '1 1 ..''. . - 1\ \ . ■ .\1 ') n m.Vaw. ■,!. .1- .t..t>',;t 

EStirireHlHetCHiHi/imiilimfetetiaib.- 
JttiHitiHmliHetffirAtir a'eib'J-iV-» . 
■ titHriifar.idm a'b inc^&tre&iftrftn*. "' 

dicHLaic&^^HMiUlUHHtjHe^fialftrkfHn 
' &*ii db-fkrUt^y cmni' dinmetertcd^'^ -. j 
•>'«f Wir«iri iit/eiHkll» iv.fr» 'i(cij.xiti. \ 
-.^iti ffatiiim fHitftiiffiialiii^reitlt^ 'i 
^iimfrtiti^Hm^HdfiiBJfjhiM^iibi 
''.tffevtitreiie.cpiHd.rtiiHmcA.. i..- , . ■ v. 

> iiim»tmnii'ah^HidHisfmiam»HSiKl.iii3StcfHiHht^ 
- ftrhfHimih hi^iiifimitffinlitya-nii/tf^.iii ftrAili, 
^'ifJInt^iMilHtn o^mdiJbtMS . v > .v ,;\ ■ , , . , ^ 

-^i 'Imi -1 MiHt ccHdb i rHtimitmluiu.timfifitjoHtxyii. 

> tHlitm fimilktmttnnHi c udaki i)r ex rHtiiHt. H»ffi/i. 
rnm , fiHi.l^ jHid idim ifi li tiniim ffir^im) ix ,tdtii»i 
ttmfirtim , Htmfe reetH c b W b h . Etri htchs c c , dd 
hi, efvlrectuigHlHm ich, odrecLagHlHm lbb,fiHevlrt. 

■iM€^.,iidbl,iifitniiHm. HimHlmidi. iriim,HreHtceuul 
1 ««<»<d,t/««f*rhi ddn^mtiii^^tKmidifi^t^t 

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wbU$f$i4jm€ftfm9$ctumh. Et^mtmisfimiAsrctiS^fifiiffMit 
fpirdlh\dd0mttesfimttlr€€t4s.yttitt^'^fttr4k§Um^ tntmsm 

HHHS drcMs c c ^dsimmdmrtctdm cd « .^^dcrdS &c. 

Siqidf ergoponatrrdatn w^^alm^U^iciivk^^ 
.crtffsfabola ddk aKjittlis^aiiailW Quiiqiiea^hy^^js 
iRodis fpaoUnrl^ii^lis,Iinef » in pa»bolam oi^siopa^turjquap 
quam non omnes per cuma Indiutfibilia pcoc edant «Bc Toieo* 
"^onaconcordaraUna^.dclifiei&^xUb)^ 4^QcbinQedi$« 






Exempliifti IX« 



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Sspiidmi^kdrimmxhc^ cMtms dxis 

bd • coBMstier^mfcrifitt^ibc. Di-^ ^ 

co^ hcmifphdritfm ipfiMsconi cffedMflttm . 

StttMdturittrtctd ^ftSdctMm^fmdtAf 

i ; fcr qModtrdftdt eircMlMs n o in htfnif .A. 

fhdrio crectMs dddxoms drfifdrficisivjfr 

JiMdricd{ihiJMC0M0>circddxemp6. . 

tMoie^ /i^» ; circttltts n 0| W- i h^ r* if/ ^M^Jfmm d p ^^ p i : c^ 

WflPf 7 ' diMidendo^ ^drmiUAcirctddrisrCttiMsldtitMdoj^ii ddcSrcMltgn 

II , \ Ainr n i ofiMC^ ip , fqMdleeftretftdttgulo (h {\tuuthter ^fextif 

ziddiiyeH Mth idiib.) £fgo ftdri^^Ai^ dd^ircMlMm ih 

ithfi di critfjtt roctdttgMlMm f h » std^MddtMttm \i^%Jm£Mt cylvtdrifd 

f9li:Jfb.fgpirfl,igs iihiyddeMMdemcircMlMm ib . A^Mldes crgofMMt 

- dttmiUdcirctddrisyCtttMsldtitttdd ni, (^fuferfmes cj^Utodricd 

. ^ f i h J«< & hcfeMtfer^uhicttitti.fitftmcSMtMxt. JSrgo oMitresfiomt 

Mrmillfj nemfefolidttm hdtntffhdticsm excdttdtMm dcmft^ cotto 

«)><^> tqMidesntmhdmtdhMsfittmlftsferftcttifMstyUmdrifis^ih 

. feifficono^hc. Proft€r€dCottiMmg€Md^jfMes.h€mtiffhmriMm 

infcrifticotBidMflMtmeffi. ^Modcjrc 



QZ 



.Exempknn X> 
rodtiiheSmistsuftpmi€mumffhierifCMm;ihc% msitittdleeft 
comoidi cmidmmUffirhoUciO e d i: edH^em disismdime 

bd hd* 







4«^ 

Um ' twudh fMfafichi ftgmtmH ^c$ii^i^ 

tmtc : cHiMS^ldU^ Mnfmmfit^ g ^fciUctt . 

€^0fenp^terjfit€tmmfi^^ 

MMmffhdijrde • , ^ ;^' \ / 

'^0mh^mdtMt.tMfdgittd h^qmtdtns^ 

fM^mMm. kjferjfmdtr^^ijUffhdmcd 

fMferfttitSfQrfCx^ frierisc^maemri^dim, 

fegmemto ,^ circMtMs^mMSTddiMs fi m» 

hmfifdrdUdMsiMcoMoide^: ^ - . 

Erstqs^mdfMferfecies z^hc^jidimr^^ : i r; j .. x , ; . > 

iir^;»i onv ^mtcir, mlms em.rddi^.^ bi^\ytd^itiMMm #«ri«^><0^9 . 
§kde^imdUtd$em zfimemre^ddemsMim-zh^ ddon^mel mt reUdm^ 
^uh^, irbd> ddtt&m^Um^huA^fimm^ ^fifidmfUs^ mtrem 
isdii^Uifn g bd;,^ g ivd ^fime. (eh bfferbHMm ) ms qmsukM^ 
h ivddquddnMim' ntjp \fi$U4^ciremieksrddie hf^dd ewftdmm 
s$t^4diei f) m ;J SeddntecedwtidfmmdeefmmUd ferfiiffefitiet^ . 
Wge^d9fddlis^eriifiif4rficses:€i^kk6 Ar, eiremk emms^wddiiil ^ 
n m * Et hofjfimfer ,* erge omm.es.ommitmsi; meMOfeffhderdefej^ . 
mentum minsis aequdlecrk comeidt hyfenhdicm, j^od efdM^di^ . 
< ^ QuaDda vero fcjitncnnimfph^riefis^ bemifpinri^ 

|^ficidlfiieimif{di^i},^ajtua^ ^ .'. 

* ^ QiMndo v^ro i^xxiiic^vocimxstfphttm^ fDaiu^ j tune. oftefl^ - 
^mr a^qual^ duobus folidis ncmpe f ruftocwdam redlo conoi* 
d(6 hy perbolici, ciirus maior bafis ftt «qualis curuat fuperficict 
^" gmenti /phs^rici , latus. verilun & f xceifus fagittse fegmeim : 
pra radium fpha^, altitudo v^ cx<.eflus diametri fuprafa- 
_ ctam «^ Etronpi^idam > fuper miilorili^kfi prsdidi fhmi co- 
Ititiukf » (TQm altitiidine , quae nt oequalis lateri verf oipfius fi-ufti • . 
Facilis demOhftratip 6ft':iqua»iqiiam pM^fitio<li£cilis videa- - 
tiir • >.»* ^ . 

Concorddn/id;frdecedentssdememSfdiiemisf . ^ 

emmJtSrimdudi^hkmsbs^. 



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qwd oftnfum €p dqkdtmht^ijl^i^em 

fhrind Ar^himt^y Vl^^ S^ frsdiG^ 
fijgmemto/fhdrico^c^^^^^^^t.^ I>r 'r.v/. -^^ix^tv 
mM' % ' PfodncdfMT i WfjmfUi^imdhf^h^ 
^h^oU: tritqiftgmemtmm mimm^z ^Cyml cmmmf' .^\\ 

*• • fcfqmiMlteidiffims g d • £r//.^; cmoid^^ * ^ ^ v^w | ^i 

cd f , mtcommm cd f, a/i««iJUib/i)(^ \ /. ^ c^a , ^v ^ Jy . ^ ,. 
.^mmifogmt mm i mjfi hthimm^liu l!^ ^ \^'*^\^\':. <) 
yiavMr a \i^\tS.vt:^d. ddd^t mmmsMtum Sih c^ml^mifsfmi 
mlimm , e.d /• . ^«at qmmdjrMtmk a^ y^tdjfmmkMmm e b ; jfinr ju( 

/^(&ciM^.abc*A/i:MMr« ad, ^ ih ifime^fiam^ 

fikxttrmituitmkir^St^ MtKhmdh.^. Kumfe m 

C if f udts yJtdnmd$m€fktm»:.fid£^ Jtlcfmmum f^gi fffftfotim 
^kMm i ^^JjfimCekmdz t/idm Mc doSiimi ArchitmcUt . 
•*': liJSmiSfSiBam mJ^ih^ 

afiptaSdx^^ . Schoc psttet ^ : Hdxnw m «f conftat ex 4«3afaus fi^int» 
diatneiris, & ex ipfaj^ df iicd jt i^.amftaf ex viiita! femidiaoiedx)^ 
A ilbBiife W^iob oonikii^oocnu . Rdiquum niafttfefttmiefL 
-; >Eatus Rjci^Knpradi^Conoidisoon eft DeceirariunHoiiaii. 
iddqiiidemdaturiatus Verfamt & &inidi2Ui[ieiei: baiis,ieduiquis 
iUud JiequiratiiiueQiet duplum c(k iaiemyer£l» 

-^*. '• JExefnpliftn XI« , \ ' . "f</ ' ^ f. . >>^ 






-rv )ASrxylimdtrms rt&mf^ afac d.^ trmif^. X jsf £X 
/^/V a d , ^r;^// o/rr^ p e , imtcltigdtmrqmt ^ 

mblatttsdhiffkcisii^ WccyiitdmtrHi^ x ; 0\^ 

qttdtur cylimdrms doccmtdimv. £m^mi^\\ \ 
tmrdcimdt cd m i^itmmt difojfft dmf^ 

-"^^^^-^ bmt 



i&m reSldnffili c der& MnHi e { cmiift d$nr fridng.t d f (/3f/- 
^em imdgwationcyn^mfgMrdniH tJfftrfeifA) itdvtmdtnrto- 
nusy cniMs bdfisftmidiamttcrft d f » 4xis vero e d • D/V^ tdlem 
cettmm ^qMoUm effefrfdiiio cylindro cxedndto . Snmdtter enim 
in axe c d , qucduis fnncinm i , .(^/<r /)^/v trAnfedt fuferficies 
cylindricM i 1 m n , circi dxem e p infolido excaunto cjiindrioo ; 
C^ circulus cuius radius io ineo cono y qui axem habet e d . 

I^m circulns ex rjtdio d f , adcitculumex radu^ i o efivtqud 
dtdtmm dt dd i ojiue vt quddrMum dc dd ci jfiue vt reUdH'^ *• .^^ 
goilum c d e ddre^ang. li e iSed quadratum d f fonitur ditf^ ^**^- 
/irm reSianguli c d c > rr^^ q$iddratum i o duflum erit r^angu^ 
UViti^ idea aqsule rcHangulo 1 i n m . Profter^acirculus ex , 5. ^. ^b 
Tddio i o aqualis eritfufcrficieicyUndrica 1 i n m ,• &hoc fem^ /•^i tth 
fcr^bicunqifitfunifum i. Ergo omnes circulifimulyfiue cd^ 
nns cuius axis efi tdy Jtqualcs erunt omnibus fuferficiebus cy^ 
lindricisfimul yfiuefoli d$ cylindrieo excasuto a b e c d . ^cd 
etdtdrc 

Q^od autem concordet cum Euclide 1. 1 2. oftenditur • Karti 
conus ^r^,adconumeumquihabetaxem ed^ rationem ha^ 
bet compofitam ex ratione altitudinum,neii\pe redse fe ^ded^ . 
iiue re^ianguli f r^^adquadratum H^ &exratione bafium, 
nempe quadrati e d^ ad quadratum df. Ergo conus b e Cj ad 
conum cujus axis eft e d^ eft vt redangulum f ed^ ad quadra- 
tum df nempe fubduplus^ob conftrudionem \ fed idem conus 
bec fubduplus eft folidi excauati abe cdy ergo etiam ex do« 
^ina Euclidis patet fblidum cy lindricum excauatum dbecd^ 
a^qualeefleconoicuiusaxiseft €r4f»radiusver6bafis df.Scc^ 

Exemplum XIL % 

QVilibet cyUndrHs relfus -ab, cuiusaxisfit cf, aquaUs 
eH conoidifaraboUco^cuius altitudofit c d ;femtdid^ 
^^meter vero bafisfit d e , qua quidem fotentiafit aqua 
Usreifangulo ah;(^critcircu/usexradio de aqualis fuferfi- s^defi 
ciei cyUndricf ab# M/fif. 

Intel^ 




tirc/idxem c d, itavffrsdiftmm conttdes 
§ridtm . fitmff deinde indxe c d , qftomet 
fnrti^» i, ft* »pf>*^ ttdnfeM in cjtindttfit' 
ftfficies cylindrtcd il, cireidxem cff ^ 
mcomide^eircmlitSy ctiiits femididmeterp 

ih, bdfifdraUelits. 

J4m:fmf*rficieseylindned ab.- ddcylin- 

dricdm W.eBwreaanptltm ab, 4<^r*- 
«f -fc/' adn^itlnm il, >f«/ eormndemfemi^dfes, dc ^ci,//^/w 
«'<>>*• amddrdtnm de 4^ ihj nemfevtcircnlns, exrddio dc ^ 

** f «^ confequentes ; nemfe fnferfictes cyltndncd 1 1 , -ej-w 

im/«rr*//«r^/(» ih; &''<^'M''?f'"^'ZnZ 
Siumu frofteredomnescyUndric^fmdfuferfictesm^ 

eircuUs fqudes erunt . wdeUcet cyUndrus comtdt ^od&c. 

Demonftraturconcordarecum Atchimcdchoc modo. cy- 

lindrus .«^ ad conum in conoide infcriptam , raaonem habrt 

compofitamexrationcaltitudinum, nempe ex rauone / c m 

teniampartem r^,(procono infcripto,accipiocylmdrmnm 

eadem^uidem bafi, fed cum dtitodine fubinpla j & ex rauonc 

bafium;nempequadrati r^ad de, fiuequadrati ^/•ad''^- 

aangulum dh, fiue reaae cd ad dupiam cf, fiue in fubmplis 

vt tertia pars c d, ad duas tertias ipfius «/. Propterea cyhndrus 

tth , ad conum in conoide infcriptum, crit vtfc , ad duas tertias 

ipfius/f , nempe fefquialtcr . Concordat itaq;cum ^-^.dc Co- 

noid. & fph«roid. ' 

ExcBT.plum XIII. 

QVilihetconusreSfus a hc,cuiusdxisfit b d , fqudlis ejl 
ftktiroidi.qKaaxem hahear^Cynetr.feffmidtdmt- 
*^ trum hafis coni ; &fi^a d c h%fkriam in (.femidia- 
mettrffhfroidisic fotentiafitfuhdHfPdtrianguii abc. 
Comfleaturrfiian^ulum fli 1 u .• aitj. , ttbfuffofitionet», re- 



107 

ctA fcff04UjM'tMtfirectdiiigMU fl tiJt0f,metil^et^/ni. 
dtms€ty*qtimjmtftiffTpi4ieySmin€0, qmttr49^fer ih i.rA> 
cirC4i Axem b d . W- ^" 

Smmdtinidm qiedl^etftmtu {) 

i ittdtxe dc.^/w itranftdtfH- 
ferficiescjUndried imno.-<^«r 
riv/vj ittffh^oidty eitius rddiusfit 

i p . Suferfcies itdqmt eyUndried -^ j;. ^y, 

f 1 , ad cyiittdricam in^eSf vt re~ hi) /^ 

ctangMlum f[ 4din. Ncmfe, ra~ 
tionem hdht cofofitdTttex rdtiene 
fhdd im,fiite fcdd ciic^exrd' 

ticne fu *<^ io>vf/ fd*/di. — .i a; - 

trititdq.cylindricdiUddeylindriedmiQtVtrectdH^^Afcdd ^J^ ^ 
rectdng, d i c ifiue mt quddrdtMm fcddip, ttemfe vt eircttUtt 
eyLTddie (cyddcirctiliitfreyi rddio ip. Sed dMeccdeittesd<jH4 ' " 
lesfunt , ergo etidm eenfequentes : uimirumtfuferfcies cytin. 
dricd i tn n o, aqtidlis erit circult ex rddio i p. (^ hoc femferyvhi 
eMnq.fiinctum i.Ergoomnesomnihiis:hecefieonus zbc,ieqU4 
lis tritffhdroidifrdaictd.. ^oderdt&C. 

Concordarc cum Archimcde. oftendemus. N5m; conus 
4 ^ c , ad conum in hemifphxroide infcripmm , rationem habet 
compoHtam ex ratione baiium, nempe qoadrati de , ad quadra 
tum/f jvclquadrati </cadreaangulum//;fiue(cum re^n- 
gulahabeantjEqualembafim^redje i^f, adyS;fiue ^rad dh^ 
Et ex rationc altitudinum, nempc bddAdf Erit erg6 conus 
dh c, ad conum in hemifphsroide infcriptum, vtreiaarfr ad re- 
^am df nempe quadruplus . Concordatei^6cumProp.29 
dc Conoid. & fphiroid. 



Exemplum XIV. 

^Stefdrdhold,veihyferh/d,veleUiffstveierreulieireim^ 
jferemtidtCuiusdxis zh > femiUtitsreiium ac fitddd». 
2 guiet 




gml0sreff$scMm 
dxe ^h:&CQfi^ 
uiHcta bc db 
extremitdte d^ 
xisfroced4t.Stt 
> mdturidm qud^ 
libet ordindtim 
dfflicdtd dc>, 
produ&din f; 

drcomiertdtttriffdfeSHoconicdcircddxem ae; fed quddrild^ 
terum a e f c conuertdtur circd a c • Dico folidumfdiium k 
conuerfione trilinei d a e , fqudle effefi^lido c f c i h , fdSo k 
conuerfione quddriUteri a e f c , circi dxem a c remoluti • 

2V^«r ; ^iir»! a c fitfemildtus reStum , w/ quddrdtum dfflu 
cdtd de^ duflumreStdnguU aef, &ide)t dqudlereiidngu^ 
"i-pdefi ^^ ^€:f. FrofteredfCirculusyCuiusrddiusfit aCy dqudliseril 
M' y?*' f^f^^fif^^i cylindricf , ^iTie defcribitur d reCfi c f , ^/rri ^m 
ac conuersd. Et hocfemferyVbicunq;fitfun{fum e. -Er^^ 
omnes omnibus . Nemfe omnes circulifimul ^fiuefoUdttm conoi 
ddtcydqudle erit omntbus fuferficiehus cylindricis fimulfum^ 
ftis^nemfefoHdodefcrifto k quddrildtero aefc, drik dxcm 
ac canuerfo. ^upd^c. 

. Scholium. 

SI quis vero dubitet, an pra?cedens Theorema concordct 
cumProponcionibus Archimcdis, omnem dubitandi oc- 
caiionem delebunt tresfeqnentes demonftrationes • 



' Concorddntidfro Conoide fdrdholico . 

Efto conoid e s pa rabolicum dbc. Oftendit Archim. Prop 
23.de Conoid.&Sphasroid. Conoides dbc^ effe fefquiaL- 
terumconi dbc . 

Eflo 




lOp 

Eftb folidum» quale difcriptum eft d 
quadnlatero dbhg^ inpr^cedenti con^ 
Kru<%oncs quod quidem folidum in para-^ 
bola 9 ^rit cy lindrus • Sc cetur in tres par- 
tes ^qu^les tam b iS^udm etiam ^i/.Eritq; 
conus dbf ^ualis cylindro fuper eadem 
ba(i d € conftituto>fub altitudine vero dl^ 
confiderabimusq; cylindrum bunc» pro 
cli^ocono4^r. • / 

latp : cylindrus ge^ ad conum dbcy fiue ad cylitidrum eius 
vicarium» rationem habet compofitam , eicratione altitudinum 
ht^dd Id .SctK ratione bafium^nempe circuli edjzd circu- 
lum dc, fiuequadrati ^ ^, ad ^^ s fiue re A^ ^i/,adduplami^ 
cumenim^i& fit femilatus redum , erit<]uadratum s ^/^quale 
rcaangulo fub ^ ^, & dupla b h) fiuc in fubtriplis , red? IdzA. 
b:i duastert. ipfius ^^'Efterg6cyIindrus^v,adconum-<^r, 
vt ^^ad^/;nempefefquialter. Quod concludit etiam Archi- 
roedes de Conoide parabolico . 

Pro Conoide HyfcrboUcO', . 
Efto deinde conoides 
hy perbolicum dbc^ cuius • 
latus verfum bt i firq; fb 
fefquialtera ipfius bt.O^ 
ftendit Aichimedes Prop: 
2 7i de Conoid. & fphfro- 
id. quod conoides abf% 
ad conum dbc^t^wfg 
zAge. Dico etiam foli- 
dum hmnog genitum in 
Exemplo i4.adconum d 
bc efre,vty^,ad ge. 

Secenturintres partcs 
a^quales redse bg^ bn^nlp 

eritqjconus abc ?qualiscyIindrocuidam,cuiusbafis fitca- 
dcm dc^ al«iudovcr6fubtripIa,nempc gx. Atfolidum 

bmnog 




f io 

hmnog /cumnUaliud/ii;iuficxlifidnVi<|Uiflattf^ 
nus mno) ^qualeentcyHndroiuf^i^jidem&aS hg cooM- 
tuto^cumaltimdinevero i/. QQf^derfJ^iBpj^iptgr&mf^^ 
dum hmnog, quam etiamcowwi ^^*i tjto^ttam £ «0£iK 
cylindri iain di^ , eonimdem/olidoiUDi vic^rij • 

lamrfoiidum hmnog^ .adcoiHim dhc^ ratioiian habd 
compofitamexrationeaititudinum bf ad gx, &exratioae 
bafium , nempe ciicuii hg^ adcirculum ac^ (itie quadrad 
hg^ adquadramm dgi {iuere<aae bgy ad duplatnipfius ^# 
fcumertim hM. iicfemilatnsrei(^m»eritquadratum ^^^equa- 
leredUngulofiib hgy &duplaipfius ^^•^fiue^fum^s/ub. 
tripUs,vt ^AT, adduastertiasipiius,^^, velad duasteitiis 
hl. Eritergdibiidum hmn/(^gyzA<;(»)»m dhc^ vt^/ ad 
iM. Q^Qdmemento* 

HcK^ be ad 4gy ^vtln ad go^ iiue vt hn ad ik 
fiue(iniiibferquial^isyvr nn ad pit. Sumpds ergo aiU£« 
cedentiumdimidijs,€rit/f ad egy vt bny ad Mi. Sccoa^ 
ponendo, fgy ad ge^ vt i/, ad /iir, Propterea folidum 
hmnogy adconum 4^^, (quodiamoftendimuseflTevt ^^ 
ad iu) eritetiamvt fgy ad ^^. Quodprorfusde conoide 
concludit etiam Arcbimedes Prop. 2 jTdit Conoid.& fpheroid. 

Profeffnentoffhfroidaliy vel/fhdtico^ 

£fto portio fplmoidis, fiue fph^ 
rx .4^r,velmaiortVeiminoripo« 
naturq; f/^^qualis ipfi ed\ nem- 
pe dimidio axis 4 OftenditArchi* 
medesProp. 3 1 .& 3 3. de Conoi 
& fphaer. porxionem dbcy ad co* 
nu infcriptu a t «.', eflTe vt fg ^dge. 
Dkoetiamfolidum hmnogygC" 
nflim in exemplo 14. ad eundem 
conum infcripuim dicy efle vt^ 
ad ^/. Secentur in tres partes «- 
quales, redae igyhiy In. eritque 
£onus 4hc «qualis cyiindro> cuius T.. 

balis 




Bafis eadenrfitcum cona, nempe d c ; altftudo autem fubtripla , . 

nempe j;jtf. Solidumvcr^^ i5«r/>^^, quiajcomponiturexcy- 

lindro A«i^j^,&excono «i^sr^iaaqualeeritcylindrofuperca- 

dbm bafi hg conftiASR) , cukA^ttidinid^ b i. Ceftflderak^i^us 

igiturtamfolidum iS^/9sr;^^^V<fi^£tiamcopuai dkfi^xzxis^ 

ftefTeiirt cylihdriiamdidicostBmteni^folidoffifflfi vi^j^^ 

Iam:folidum hm nogyzd conum^ 4 b c, rationem habet Gfttnt 
poficam cx ratibne altitudinum ^ / ad ^ x ; & ex ratione bafium» 
nempecirculi hg ad^r,{iuequadrati ^^ ad quadratum gd^ 
fiue re^tae bg a^ duplam ipfius^ o . ( cum enim ^ /^ > fit fenula-* 
tusre<5tum, eritquadratum 4g aqualeredangulofub bgjSc 
dupl ^ go) fiue fumptis fubtriplis , vt ^ x ad duas tertias ipfius 
g^^veladduastertias ^Z*. Ergofolidum ^/Ki/i^^adconum 
Abcy erit vt / 1 ad Bf . quod memento . . 

Redla^^ad^r,eftvt /?^ ad oeyGuevt^l zd /^,fiue(1a 

fubfe/quialteris/vt // ad /iiriComponendoautem^r ad eg^ 

eritvt fuad >y/,fumptifq;antecedentiumdimidijs,erit /> ad 

r^,vt ip ad ulyiiuc adpbi Et componendo ,y^ ^dgty erit 

vt / b ad bp . Propterea , folidum hmnogyzd conum dbc 

f quod iam oft cndimus efle : vt / ^ ad b p^ ) crit etiam vt fg 

zdgc. Quod prorfus de portione fphaeroidis concludit etiam 

Archimcdes Prop. 3 1 & 3 3 . de Conoid. & fpha^roidibus » . 

Pluraadhucexhibere poteram excmpla demonftrationum 
per Indiuifibilia curua procedentium , nifi fuperflua , immo etia 
& molefta exiftimaflem.. Hoc vnum admoneo ledorem,in ma-. 
gna pjirte pra?cedenrium Theorematum me fecilitatis graria fe- 
cifle cafum Propofirionis parricularem , cum tamen facere po- 
ftiiflem vniuerfalifllmum. Exempli caufa . Poteram ^in figura 
primi exempli^ fupponere tangentem b c cuiufcunq; longitudi- 
nis, & deinde oftendere ita efle circul .im ad triangulum , v t pe- 
ripharria ad tangentem : fcd faciliorem conclufionem iudicaui 
aequalitatem interre, quam proporrionalitatem jprxferrim cum 
infolido Hyperbolico de a^qualitate tantiim ratio habeatur • 
Si itaq,- Corollaria limitata plerumq; demonftraui , vice Theo- 
^^^t!!^9t^^"^ , fcias data opera fa^um efle • 



ItS 



l>e6md<yl 



tMimrJoUdimfitt (Jiftcundtm^ucem confidcretm) Unfitmdi^ 

iit infinittm^^ti$d^MHlcmMtttjm 



'f 



PB 5<S ^ 



. "■ '■■■ xO 

DE SPLIDO 

Hiperbolico Acuto • 



LcmmaPrimum . 



iSTO hyferh^U^aiiitsdJym ■ 

\ fWi^ntAhyHc, difgu.- L 

C l$tm re^Ktn ctntinetKtt ; ' ■ 

k t^ retitht^ fS*^^ ^*"^ ■ '^ 

I dxetn zhy fdcifitHfifff*^ 
•• M/UmrfhMthndtntimhy^:' 
ferboUetminfmiteUitguverjmi hi^fum^: 
^idmodumdefnitttmeH . InteUigdtur iim 
intriifptm deutiimfoUdiim\Ti^4ngiUum 
MiaJi»dferdxemibdu^um,futd defg. l>ic» hic ritifdMi^ 
fmUtmdqfidlee^eqmddrdtofeitddxiiiffiushyfetM4.\' '"' 
HmcdtMr ex a eemtr9^f«rb»U,fim^dxis aiv, qmimigulttm 
b a c ii/dridmfecdtit \fdtqi reHdng. a i h c ; qm§d»mnii$h qmd- 
ehdimm erit (ndm cum reSdnguU fgmrdjUj dngmUs a ijGf- 
ridmdhdxr^h dimiditimy^. rrgi quddrdtum re£fd ah di^ 
flmm ttit qmddrdti aibc^ fme dufUmreifdngmli af. &i4e% *tufii 
fqmdlere^dngmlt dcfg, S^tderdtfrefefitmm^» *mm4tQ9^ 

Lcmma IL 

OJSnfs^yUndn., eirei eemunemd- 
xemimtrifelidmmMittimhyftr ■ 
holiemm defcrifti tifeferimetHfiimt. > 

imtttigefemferfinebdfihiti.Eiiedem^< - , ^ 

Mmrf<itid»m,cmiusdxif ab, (^imtrd 
iffMmimtetligdmtmrd^crifti ctrciet' 
mnttem dxem ab fuattshet eyUmdyi . 
Cdef, ^hii, £rmtttqi*fMdItdre£(di 

- / ^mU 



%P4^ 7>e SoUdo JfyfirMici 

. A,.. fMUper^xem ccgl.erg» ^qMdlcs erunt etidm c»M4 cyUmini. 
aonicrMtiaitpvf^"- ^</*r^. " X * ' i '^ 

«* /^*- Lemma IIL 

OMnesifeperimetrhyUndrT(cmMtm»dfpmtiRi^Miimdea 
tefiUd« hyferhtic» eUfitihMf^MH) interfi/mnt ^did. 
metri/MMrMm bd^Mm . ^oniMm emm , inpMcedettt» figmri ^s- 
qMdlidfintre&MngMU zczUeritVt £c dd i\, ttm izm4 
af. IdmcylindrMs c€ ddcyUnthtMm%\yTdtienembdbeteem 
feJttdmexrdtienedMddrdtiH ddqMddrdtmft aii ^exrdtf^ 
mereaf fe dd il ijime ex rdtienereffd iz dd zf, velfMd. 
drdti ia ddreffdmgMlmm iz(,fT»fif eredcyUmdrms cc ^cy^ 
litufrMm £ \^eritvtqmd<kdtMm, fa ddre&dmgMlMm^^ f; -«* 
feytfc&d fa 4^41. ^idi&c» : 



t 

Lemma IV* 



»\ 






ESf fiMMim Mmtrn abc, ir«M(# 
4W ^h,%drefMtrmmhyferMdpt< 
fnmSMm ^AnitufiiUcet djymftdticdm 
meninne . didfdMtemhjfperhplf/itdtJ^ .v 
tdtiigitMr imficnm d, dd tuterMdUlMt^ 
' Al^e^riftM/fhdrd aefc, ^ »4J<frt 
> IW4 m/ em^M» intrd dcmtSf^UdMm dt' 
' firiftihiUmmexcentre d. SMmfte^icy- 
/indr» ^MOcMnqMe intrd dCMtumfiUdMm 
defiriftotfntd gihl. l>ic» cylimdrl gh /t^fcimfi^ffi' 
drMf Um ejpfif er/iciei/f h^rd 9lc£c^. . 

CMm euim reHdmgMUm gh^er dXemcyUmeki* fqudep 

'^ ig l^qMddrdt» df, eritcyUmdricdfMfer/UiesdqMdiiscirtMl» fMifit 

^,\ exrddiit d f memfecircMl0z.cic^ .Preftend tddm/Mfcrfciet 

cylimtkicd -g^ihl /MhqmddrMfU eritfiffWfieiei ffhdrd aefc. 




i • - 



ic/MbfMddruflmsiefi <» ^igtdd^ 



\ • 



«» 



'\ 



Ixm- 



Lcauna T» 

Cria^nfiltejUaSn ghil Mm/ilubimdeiaiimtUfmf 
tiimaiftfctdeittifginiJpitfrftiisfiiKidfiitstiliu- 
UsefieireiilteliiilsftmidUiiuttrfitUiieit df. memftjemidxist 
fitiefimeilMtverfiimiffiiis kf/erhlil- Hee emmi» ifftfrt' 
ftejfnfrieeieiuislemmdtisdemtiifinuiimefi. 

Theorema* 

SOIidiimiciitiimhypcrbolicuiniiifinii^Iangutn,lcaumpIa- 
ooadiucemcrcao,rnicumcylindrofuxbalis,an]iuleeft 
<c:yIiiKlrocnidamreao,cunuba&diamcteriii:Uiiisvetfam,fi- 
ue axis hyperbolc, ahinido vcr6 iit acqualis lemidiamctco bi; 
lis iplius acuti folidi . 

Efto hyperbola cuius alymptoci 
dhi de angulumrcdumcontineant; 
fiunptoqsiohypobolaquoiibnpun- 
Aod, iixxmdt asjaidifiaas ipfi 
ditlt dftea^iaHiWSde.latvaf 
iiel%aturvniurrrafiguiuckcaa]t£i(j. 
fcd vt fiatfolidum acutwn byperboli- 
ciim rji/,vnaciimcyhndrofuieba- 
&sfede. Poducatur id'mB .itaTt . 
ii4;a]uali> Jit inKRK> axi , fiae latoi 
r^o-ny^«rbolx.Etcircadiuneixiiin. ■■',,'■ 
Jth iatclligaiurcircnliii eteat aJ ai yiBpiu t u u dr: ic&fKt 
bafi <«i(concipiamrcylindcusrr^Lii<<f^A, euiusaldfiido fit 
<(c,nenipefcinidiamcterbafis'acuiifol<di.Dico foUdum vni- 
.. uerfum/r ji/f ,quanquaiBfinefine(oiigmni>qiialetaiiicnef- 
fccyUndro^Cj}*'. ■ ni.: 

AccipiaturiarA&aiair. qnodlib^pnidini A & per t. io- 
tdUgAur du^ fuperficicscylindilca inti infoUdoaano 
P » cot»- 



compnthenia circa axem dti item.ciitulus im m C)rliiidio 
^cgh «quidiftans bafi dfh.': ^ ; l ^ 

Erit ergo prasdida fuperficies cylindrica cmli zd circuiuii 
im] vt fe<9anguluni perwcm .^Ji ad^juadraram radij circu- 
U irMi;iiemp€vtredanguium «/» adquadrattnn femiaxis by« 
perbolse i & ideo aK]ualis ex lemmjue . £t hck: femper venim 
erit|Vbicunq;fumaturpun(S»mi. Prcpterea oomes finaul lu* 
perficiescyltndrica&hoccftipittnifolidEum^cu rAi/, vmu 
cum cylindro h^ds/e dc^ asquale erit omnibus drculis fknuii 
hoc eft cyiindro ^ ^^ ^ • Quoderat&c 



SchoUum, 

4 • -• . 

.' 1 - - » . ; . /, • l . . . . . , , , . J l 

* ' Ihdtkiibite Videri poceft > citth iblidum hoc infinitamL longs 
tndihf^ tiabeat^ nullam tamen ex iilis fuperfidebus iyim- 
dricis quas nos confideramuS|infinitam longitudioein babere 
fed vnamquamq; eflfetenhinat^; vtvnicuiq^patebit>caivd 
modice £[miliaris:fit do^brina Conicomm . 

Verit^m .prsrcedentis Theorematis fuis per fe clararo , & 
per g^<m{iia aa.ifil^um lib^l propbi^ confirmatam izs^ fn- 
pc rq;[yt d . Taihe^ vtlh hac parte fatftsEbciam le&ori etiaa^ra- 
diumUirisii p^runTa!T9ca,l itdrabo :hanc ip&m demonftrado* 
nemiqlcajceoperis^^iperfdUrom veterum Ceometrarum viam 
demotiftrandit longiorem^qdidem) ied ndn uieo mihicertiore* 
, Int^in|i) <iuxa demonftcatiohei okliibcbimus de illa taittum 
acuto fdlidolcaius hyperbbbegenMSrids afympcoti angiiiu&i re» 
Aum contineant , dicamushici>bit& ^ omifsa dhEtiooftrAticvie » 
iQi|iM]S' figuris irmiaftirfiiit acuuifoiidai .quaiido^ftfyinpfiotoa 
^ngulus obmfus nibrk » vd acutus • ' 

- i 1 Dmi^nftrMhnes ^ qti^JiitmtMnimim^kmfrfUfimms ^JHi 
4eSw imim&fiMi^teiUjHgi^ . ^ 

^ £fl o hyperbola cuius afymptod dh\ dc vangulum obtQ^ii 

xoimxafiiWseaf^^ fiacibUdiKDAcu^ 

A ^ fjr' * : «» 



I » ^ 



ttjf 




PfvUimSecMffdtimi 

t&infinit^Ion- 
guin veifus^.% 
lcccmrq;(vtin 
prima n^Jph 
no rf* ad axe 
cre(ao.Eritio- 
lidum acututn 

lindro tfiUy 

&cono /rf/. Infecunda vn:6%uraiitplaiiumfccam 4<r.erit 
iblidum acutum vniuerfum quod imponitur fupcr circuIoVr 
fumpcoetiamcono #i««,aequalecy]iadro ie» &cono />r 6.- 
mul fumptis . 

Quandover6anguIus . - 

afymptoton acutus pona - u^ 

tur, &ficplanum fecans 
r*/ inprfmaiiguija.-Brir 
folidum acutum chdvm 
cumcono eai sequale^ 
cylindro tf<iV. Atinfc 
cunda figura eritvniuo'* > 
fum folidum acutum fa- 
^um ex conuertfone quadrilinetnincti ^ti^iiA fiKfltai^iMgi * 
duplumcylindri /««/^. . j , I, j . .1 v. ■■. ,,m 

Sequuntur idm fub nomine Corolldriortukt PropQ^tionAS 
quadamexprxcedennThcorcniite prbnumSies^qiwqiude^ 
a!iqiy$tfn-an-ogan'qas Wusacutifolicliliy^rbolid lian^cOA 
coiHcmnendasdei[nQnfirabua(.-'^ .'-..! 'v _., \,. 

• '^ CofoUarium Primum. ■ j •: ; 

' \Afiitd/»Uddhyfeth*licd ebd, n bl, ^tMhtjfgKrdpdg^itsi 

txftBiemik. c d > n 1 dd dxtiftiiis ^tintfVmd^itmcyUmdtitfmA 

tnmi bdfimmi , imteffefitmt vt diMtetri tdHtttdetm kdfimm , Mtmtfn 

.v/tre^dcAddrsi.- ■ \.-' ..i .,v..v 

Hmrefimftdfrfetdmiis Thewe,mdtt»fiffirdr& cnt^i^ 

ifieat 




„1 DiSMtBfinMi 

ekme^mtfilidum fchdc^£^dlefyB»A0 z,cA. &fi^ 
dmm onbli fqitde cytimdt» aitnh. Erg» ^ii»m ,4d f»b^ 

dmmentveeyttHinu^cjUiiintni.Hemfevt cz, 4d **».^ 
/kmftis ib^Ue vtweifd f c dtCo i ifiue v^ c d 4/ ttl - Sgfd 
erse&e. 

CoroUariuin II» 

Aentd /iUdd fyfer6»ticd d b e , h 
6 t » etidmfine eyUmdinr/kdnlm bd^ 
fmmfimftd, imtewfifimt vt dimmetri. 
edrmmdent hd^mn , ttemfe vt de dd 
hl, Befiriftis rnim bdfirnn eyUndris 
cdc£s ghit,erittefttmfelidMmcd 
bef» 4fdt«f»mfiUdnm g f tb I i v/ c f 
dd%i» 'SedabLune cyUndriis cc dd 
sbUtnmeyUndmm %iejlvt .c£ dd^i, Mtpreliqnnm etidm 
.filiddm &bc, ddrtUfmnm bbi eritvtte^nfnddtetnmi mem' 
fevfcfddgi, Hecejlvedcdd^hi, ^ed&e^ 

..» • - -^ 

CoroUatium IIL 




CCLAIlt 



j i 



fUmis abycd,ef»gp 
ki vifi&jufmim fhmdUmetri 

mitdtf€froff^diU€T(^$dfMiit 
fn ^fidKCiftdddidUbimm i 1 , f- 
^MdUisiffiilfeceMtmrlvRjaxn, — ^^«r t 
Ito &c. dMaisq; X^tat ^ ,OW«WtI 
VLC&t.dtdiM&emfdtndlelis^ferfmnBd g,<^ f^&c&s.d^mt* 
tmr/ecMtidtfldmdt.) JOictommidtfrmstdimterceftd dqmddidtefi 
trnmimterfestmmetidmemcmtpfilidm-^VLl^ ^ \ . \ . . • 

rMtethoCn NdmfrmmdiCMtdt/hiiddifmtvtdidOffgtwilid^Mtm& 
imtlm;^nfMdtMm€ilik4fiMmfcmamtmr ^iit mttmtcrimi^Meii^ tk 




v>titMtfrtgrtJiiiitet,etirii»Mia<ifiliJJ $ap,eu{, cudrf». 
ia cAdrm Mimctitl tttitnt tmtu . Ergl tmntt txtljpltjitm- 
fe^ctimiAffnSttqiuiittniitMiituirJi^iumitiMt^tittifi- 
iiHt ^uf.vterJltfriftJitim&t. 

SehiUimi. 
Potent ttiam proponihocniodo. Sifiieritfolidum acutD 
feaum plano g f vbicunque. Sumaturq. b f fcmiflis axis 
*».t>eindefumaturjrier.parsaiusf»iiteruniqiaccipiaiurr/ 
quar.t>arsaxisr/:pofteaaccipiatur quintaparsrcliquJaxis. & 
hocfcinperi&petpunaaftr&onumplanaaganturietuftt ead, 
luiupm&c. 

Corollarium IV. 

AeiiiitiiifiUibmkyf^iiUeimtifiiftiimfliiwdnMirt 

Itt^tlqluditHtjliiidrlfii'^'^!- , ' 

EftifiUdnmtciiimt abc tifciJfmBfUiu ac tddxtm irt- 
iti(hic nim midi iniiiigemm fcmftrflMaftcJMU, Jiud 
ctirtttmeminiffiX^miiumfilidiimiiifmtifridMamiidd 
flt,tlt\,.D«lftiidtlinsAc,ijiitlttJfityUmlnfmtUfiimmi 

ffA^ce: V ■ ' ■ ' 

rititiiimeyUiulriu fei%vi inTlitirt— 
mtttt^fdg- 1 1 s.EntptitmiijiUdim d a bc e, 
txdtminpdaitfqiuleefUiidrt !i. iMly. 
Undrns i'\ , tdcyUndrnm & c , tiuimtm hd- 
hettimficfitmitxrdlilnieimulrttiittdd {e 
tS- ixruilntrtllt {cddcci Jnt ijlliduti 
{c ndrianngmlnm (cc. CyUndrni ittqi {i 
tdcyUndrnm dc, tSvlqiudrtum fh td 
natngidnm fc , mimfidnfUs. pnflent 
fiUdmmVmiutrftm iibct (imm tqmtU fil 
tyUmdn {i)JmfUimttittyUmJri Ac.Btdi- 
Mifim,iru/ i6 iU imMm» ibctftmUfmthtlifiyUMirtitct. 

iigttl&t. .,/.».,.. .. ■ _ ■, " 

Corok 



ito 



HyfeiMkk 




Coroilariain V* 

\AMfmfim htmUffhdrimm f b e , Mk 
tfdfitldum dCMium infcriptibile ex d 
€enm hyferb^Uy fubjifquidlterum efi 
vuiuerji/olidi fh a c c ipfum hemiffhd^ 
rium dmbieutis . S$lidum.4u$em f h a c e 
€0uSdiexacutoJolidoinfmitelfinio ha 
c^(jrexcylindrobdJishemiffhdri$imtdM 
gente fhcc. 

FdSto enim eylindro ie vtin Theore^ 
mdte fdg.t is^ erit hemiffhfrium f b e 
fubjtfquidlterum cylindri i c i Cum edn^ 
dem dltitudinem hdheaiy dr bt^fim edndem , nemfe tircnlum cn^ 
iusrddius eBfemidxis d b . Subfefquidlterum ergo erit ifjum 
hdmijfharium etidmjolidi fhace, quod fqudU dtmm^rdtum 
eficylindro ic^d^. 

^ CoroUariam VL 

XJloJoiidumdcutum cninsdxis Sib, (wfgmdfdg. tit.)fe. 
0umvhicnnqifldno d e . Secetnrvno&dlterofldno hUquod 
Cdfidtfortionem dxis dufUm . DicofruJtumfoUdum dh 1 Ct i 

fecdntibifSfldnisiMerceftumsqudle(pfoiid«dcnt0 hbl//*i 

fuferimfojito. , 

Citm enimrefidnptld c c , g I fntsqndlid^.&ldterd arum 
recifrocd, eritreftu d e dufUiffws hl , &idiofolidnm dduti 

^bcdufUtheritdcutifoHdihhl^&^disiidtndo^finfinm diilc 
tqudlc ertt dcmofokdd hb 1 * J^^^a 



- 1 \ 



« t 



» •■ ^ 






. • t \ 



i • » 



Hinc manifeftum efl^ qu6d fi acutam 
dum cft , fruftum interceptum d6 U ^quod duas bafe$ haiK^ 
bit; a^qualerempercritcylindfo minoriibafe ^iJ//. Subdu- 
pluffl Qero erit cylindri maions bafis t def. 



Koblema Secundam t%i 

CoroUarium VIL 

tt/jfUtM/ ab> cdy eU fecdmti- 

bus MxemfiUdifrtftrtitmiUtefihoc 

eSttJtivt gh A/gi, itd gi dd 

gl.DifgfiitJfiim zcdb adfrmfim 

cefd effevt Wdd ih. nemfei» * 

r€C$fr9C4 fdtione dUitnMnMm, 

CmmtnimttSidnrMU gf, gd, 
g b . fint fqmdUd , & latetd eormm rtcifncit , trmttt trts^eBm 
hb, id, \ii,.inesdemc^ntinmdfr»fortMneim^nifmnt%\^ 
gi, gh. SedfeUddiantd aLoh^codt eQ( fmmtvtbdffmm/e* 
mietidmetri hb> id, If, fimevt gl, gi, gh, ergo exceffms 
foUdermm interfe ermnt vt exceffms Unearmpt . NemfefrmBmm 
folxdmm icdbtddJrmSfmmcefd eritvt \idd ih. ^mtd&e, 

SeheUKm^ 

Ex demonflratis patet prini6 , quomodo datum fivflum ^ f/^M^Fj^ij 
t fecaripoflitplanocj/jitavtfa^ponionesinterfeiint vtal- «^ 
utudines,reciproceta[nenfumptz. Quodquidemfitfumendo 
^i mediamproportionaleminter r/,^iS. 

Maoifeflium etiam eft, qaod fi wma 
turquodlibet {egmctumaxi$,puta di, 
&|fecetur bifariam in c . deinde .« f fc- 
cetur biiariam in d-, reliquum autem 
4<^ bilccecurin f^&ficfempcr. Erunt 
fi-uflafolidaintercepta^planisper t, 
Cfd,ej du£tis,incontinuaproportio 
ne in qua axes , fiue aidum d^erenti;. 
£ritq;primum,&fubtilifliaiu fcuRu/i 
a»]uale acuto folido fibi fuperimpofito . At fecuodum fruflu du 
plum crit primij tcttiiiquadrupUim primi^^. verd o^[ddm,quiQ 




1%% DeSbUdoHypedJoIito^ 

mm fedeeupluin : & (ic ieniper i quo magis ad centrum ^ acce 
demus^maiora praxedcntibus eruatfru^» & multiplicxa feciio- 
dum numeros in proportione dupta progredientes ab vtiitate« 

Siverofamaturquodlibetfeginentumaxis 4^^ cuiusilup^ 
lumponatur dd; &:ipfius dd duplutn fecetur 4c^6cGcde^ 
inceps s eadem euenient ^ vt fuprii di£)mi cft • 

QB^cunq; autem diximus cxdBpio allato de ratione dvtpla » 
verum etiam eft detripla > quadrupla ; fefquialcera « & de qua- 
cunq;aliaratione« 



Corollarium VIII. 

iifoMumdcMtmnfti^imfut^ 
ruflMifSibjcdt ef, gh^c^r. 
iidvtdxisf^ftiM€S icctrc i m^ 
cifieMts^nimpclXy Im,^ mn» 
n o ^c. dqmdlesjint i tritprimik 
fruJlMm a d , /d ficnndum c f 
vt^. ddvnum : fecuntbtm wro 
Jfrujfum dd tertiu erit vt^ddj ; 
Tertium ddqudrtum erit vt s^dd 

S • qudrtum d^fuintum vt d. dd 4^ ; Bt sicfemfcr vt numeri bi^ 
mdrio differentes ; ddditdfiilicetfemfer vnitdtevtrique termi^ 
mordtionis. 

Ndmfolidumdtutum a u b ddfotidum dtutum CMd^^eHvt 
al dd cxvi^nemfevt miydd iXyboceJiduplum^ Etdiuidem^ 
da , eritfruBum a d fcfudle folido dcuta c u d . fue vt 3 . ddj . 
Solidum vero c ud ddfoUdum cuf ejfvt cm dd e n yfutvt 
nidd imy nempe vti^ddz. Btfer conuerfont rdtionis tritfo^ 
UdumcwA ddfrujium ci vt i.ddvnum • Ergo tx fquotrit 
frufium ad ddfrufium civt i.ddunum . ^oddrc. 

Eodem modopenitus rdtio reliquorumjruftorum conftqutmi^ 
$ium ofiendiiut efc tdlis qudlis propofitd eSt ^ 

SchoUtm ^ 

Patci in progrefTtt demoaftraxionisprimutn fiuihioi di a»^ 

ouaJe 




ProbkmaSecundum. xx^ 

qualeeflefolidoacutonbiimpoficoriri/. i^iecundumfruftu 
r/*<iu|$[iimeftiolidi r«/iibiim|>ofici;TciiUKa ver6 tripluau 
quanuin quadmplum . & fic in iafiaituin . 

CorolUrium IX. 

Siftlidum acntim i eyUmdruisfmftrfieitbms dimifmmfiitrh l 
tntntfeUdd MnnmtmMimter eylimdrieds fmftrfeies imttretfttd « 
mter fe , vtfmnt f<irtioatf djymftmmh iffis cjlimttrieis/mforfi- 
eiehms abfci£f . 

Sitbypth^U^hCj t^ timem qmttcmmqi ad> bc> cf.fd- 
rmllcU afymftm hii^efmmtrtdtmrjtgmrdcireamfymptot» bi. 
DicafoUdmmdefiriftmm i ^madriUttto ebcf, adfoiidmm dt» 
fcriftumiqmmdriUneo dabe, tffivtreHd £e dd cd, 

Fidtenimeytindrms \i ■, itt inT heo^ 
rtmatefdg.i I j.eritq;foUditm nmicf 
m^mdefyUndro I i.EtfiUdmm poibe» 
mqmale cyUndro 1 e . abUtis trgo tqmaU 
bms fremanebit cyUndrms i£ fqmaUsfi- 
Udofibi reffondenti/aiio a qmadrilint» 
e bc f. ParirationeeyUndrms ce (qmO' 
lisojitndttur feUdo fibi re/fondentifd' 
Ho i qmadriUneo dabej eritigititrob 
mquaUtatem ifiUdmmqmadriUnei ebcf 
adfoUdmm quadriUnei d a b e vtcyUn 
drms {£ adcyUndrmm xz^nemfevtrt- 
^a£eaded* £tmod&c, 

CoroUarium X. 

AcmtafiUda abc, At£. fmftr bafibmtaqmaUbmt ac, df 
comfittmta , ^ a conmerfione indqmalimm byferboUrum defcrip- 
tayfmntinterfeiudmfUcataratiome aximmfmarmm hyferbola- 
rmm, 

IntelUgatotmr «nimfmb baftbHs fiUdtrmm cyUndri bc, I( 
.^ g^ % ffit 




X %4 De Solido Hyperbolico 

eruq fiUdum 
zhc dquMlecj 
lindro hc; & 
foUdum def. 
Squale cylindn 
If. Profterek 
/olidu ^bc ud ^ 
folidum def» 
urif vf cjfliud, 
hc ud oyliud. 

\i^fiue(cumuqff^l^shdf€s"habeunt)vtdntudo h^adulritu^ 
dinem I d . fi^^ ^/ re&dnptlum ho4d reffdngulum 1 f . Soc 
^fiif^^^pisi^ndihus^vtquddrdtumdxis ln ddquddtdtstm 
^xis^mo. ^9d^. * 

CoroUarhim XL 

^CMtdfiiidA 3.\>c, def 
fiffd 4b inaqMdibiis hyfer- 
b9lisy&feif4fUms ac,df 
itdvt fmionesAxis Ih, oi 
fqttalesjtffti erunt inttr fe 
vt bafes , nempevt circuUts 
ac adcircMltmdif. 

Hccdutemfdtet. Ndm 
filitium a b c dqudle eB cy 

Hndto cMitts bdftsftt a c dUitttdit t)^^ 1 h . &foltditm d c f ^- 
f *4/^ tfl tyUndro cttius baflsjf ^ f ^titiido ^ero o i . Brgofo- 
Itditm a b c difoUium d e r <fr// v/ frddiitus cyUndrus ud di 
iiuiH cyUndrum ^nemfe ( cum squdUs dltitudines hdkedttt ) vt 
hdflszzddbdflrh^i. ^oderdt&c. 




•' . Gorollarium XIL '*^ • 
'dciUdftUddquscunqiflnt ab c, d e f i inf-fs- iidusfdgjii^ 
tfrflef^t nttf4UU rj0htng^dflquddrdto Axium ikjferbotarumi 

"^ " dlti' 



ProbreinaSec«ihtIuin.i-- ^ij 

•jtUitudinevttti duimttn idJimAnmdtmfolidttm . Htc tjl, 
fiUdMm3\>t\tdfitidim Mtru vtfilidum fMMdififtdum 
tilJiqKjdrMHlxitinijUtitmUlH a,c.fdf<trtUtltfifldiimhjlJi 
qMadrataavJt mo, dliitiidiiu H. 

faliii tnim dt mtrt lylindrit h c , 1 f i rttil tyliiiiri tlcnJ 
cjliMdrum \i camfcnttllr tx hit tribnt rdtitliiliut . ittmfttx 
ratientMludiiiiihi 4dli.& ex rdtiint tMfiS,/iiie txrdtit- 
Mir rtilf zcdd A(, iterumq; tx rdtimtrtiif zcdidf. Ergi 
rittiotylindri h c dd\i,amftnitiirexrdtiiLntTtltdnguli hac 
*d Tiadngulum Idf, tiut quddrdti i IV. dd ijuddrdtum 
mo,&tx rdtime rtSd ac ddrtlldm df. Prifttrtd ttidm 
rdtiejilididcutiibc ddfelidum dctltupi itTcemfiJiideritex 
rMiinequddrdti iaddmo, &.iicx0ui>cttlte ^yditelfdm 
d£. £r^lfdtetfrifij!tum . " , ' _ 

Corollarjum XI II, "i^ 

Ddtldcutiji 
lidifrujil qui- 
cunq; adcb, 
dqudlem ifficy- 
lindr» exhibere 
fufer dlterdfui 
hdfe qudcr.nque 
sit , futd ab. * 

ridtvtreBd ab dd Ac imefddl^. Sici eytindmm hb 
cuiusdltituditi, (g^idpviri ab dqudltmtji fiufie ac. 

Di.tdiur dk fdrellelddd c(. Iritq; fc,«/pe,r/ de,- 
fue Kf */ fa . prtftered o(dd (e erit vt fl( dJ^Li.Sed 
ct dd(e eg^tafdd f K , ergoferfertiiridtdm eritof dd 
fg vt fa ddiK. Sjudmemente. 

Idm dcutumfelidum amb dddeutumfoUduniitfCiSvl 

ab rilid dd icutlut l(ddAe, heiljlut i(dt{U, Xrg» 

eritfilidumdtutnm itah, fiui eylindmt \h iffi uqudlit , dd 

frnftum adcb utliddiK ,heeefto{dd(%,hee efuej. 

" ' ' , Uiidtui 



i 1(5 tk SoUdo Hyperbolico 

lifubrMs Ib AdcjUndnm bhv ComS^ipiwcylindrm» \h^u 
dem habiTe ramnem &4idfruUMm a dc b,<f adcjUndrmm bh. 
^MTf cylindrns h h ^tqu^smtddi^fiMfie^dcntiJ^Udi^ ^ /i- 
jtcrdlterdein/dembjtfi^ ^iu^di^c. 

CCoroUarium XIV, 



r Chcumfcriftus: cyUndrus a e f b ^^ 
, fiufiumacutifoUdi<z d c b . ' eHuidiu^ 

j»i^/!r . a b mmrisbasis dd diumetrum 

dcminmstdfis.ijFiut enimut zhdd 

dc //ii gh 4d hi, i (jr erit cytindrut 
. amlb fqudUsfiufidfiUdQipjer Cer.fr f 
i eedens . ^ CyUndrus^dutem . a f ^^ r//^;i. 
' 4iv«i. a 1 rJf nr/' gh i^^hii )&^ri/? 

ut ahud dc, ^^urecyUndruscircumfiriftus acfhedi 
.4dfiufium.^Cieritutreff4..^h.ad dc./^od&c. 




CCoroUarium XV^ 



_> 



l^rufiumquedUhet dcutifilidi adcb 
. iii infiriftumfihi clyndrum c dc fytfi ut 
didmeterhdfismdieris^ ahydd didmetrum 
minoris bdfis d c •- ^idt enim wahdddc^ 
t QpT^lzi itd g hdd hii eritqicyUndrus al dqud^ 
- Usfiufio z.c:iEritinfifcrcyUndrus iA ifo^ 
^•-i^i ferimcfer cyUmko cCt^udsidoquidemU^ 
^^*4fr ^^^ eorumfi£fdfintrecifrocd , & ideefre^ 
' ifdngiildferdxemdqudlid . Erit ergo (fer 
: lemmd j. huius)cy Undrus a I fiucfiukum • a d c b ^ A/ cyUu^ 
drum infiriftum e c vtiidmetribdfium;fiutvt re^d ^h di 
^cf, hocefivt 9.h dddc..^ol(Jrc. 




AEHFB 



Corol* 



Froblena Sticttttdum* i ^7 



CoroUarium XVI. 

Truflum qM{liittiie*tifoUii adcb.wf- 
iimmfnfmiondtt efliitttriiifmftilm^drcir- 
tmmfcriftHmfiUtjltsiritm. 

XlimoiiflrMimtmmtflilidlioiluptctdtlt , 
tihttsCoroU.qHidcireumfcriftus ejUtitbrittji 
e /uifiiiHimi 3.ich tftvtrtUd :i\> jidiki{. 
fruflitmvtro adcb tldmfcriftum tjUiuk^ 
tH-ut zhndAi. ErgoconStitqiiidfiilflltm 
rSi mediafrtfortioiute interdiui cyUndrBS, 
SiMoitritt^c. .:. . 

. CorollMium XVlI. 



AF 3 




Ddtnm dcuttm/fiUdMm a e b iadd^ 
ja ratiBne fccare vt fdd g. Fidt vt 
g dd iitd.d*tA\i\. Mdi\ .&ftr\ dg4- 
tKrfUBttmcA.£rit^ieoniiertend»t& 
chfitnenia f ^ g. simul ai^vtWydi 
hi#/«^wab ai cdyvel MtfoUiMm 
acb difeitdum ccdi ^Attidend» 
fdtet frofpfittim* 

Siverhbafis deuti folidi fit c^y <^tpmedttltud/ee4re ite- 
rum inferius vtrfushfperbolf centrum fldns a b » itd vtfrufiS 
acdb ddreliqnumfoUdumct^ qudmlihet datdm rationem 
iabedtut l dd^^^Ita imferatdexequemur, Fidtuti&ofi^ 
mul dd g, itd ddtd 1 h rf<J( h i ,■ ^per i ducaturplanitm a b . «•« 
queut U&%fimul ad %,ita 3ih dd cdifiuefeUdum z.Qh dd 
c e d j d" diuidendofdtetfrofofitum . ^id erdt <^e. 

CoroIIarium XVIII. 
Ddt9foliie4cutofe£itfUn» z\} . fruBnmdfcifeft cabd 
Kr/Sr-f n centrttm hyferhoU , quodfit d^ttdU cuicttmqi ddto cy^ 
Madrg g h mttis etidm imme^a . * 

ridt 



a 



12^ JHSiiS^^^ypeAo&A 

fi*ttiteylmdrju Alddfjr 
Undr0m g h itdrei^d nl W>- 
ta Mdrtiidm \ f , ^ a'/^^ f d 
duSieq;fidM9 d c . 2>ie»/rM- 
^^MmchdqifdUe^^ecyl^^dt» 

A'<«» cyUti^us d 1 W g h, 
efi.Mtre£ld nl diliyt^etn- 

mertenioyComfoneniotiterM' -£ •NJ' * *• 

^MtcanMertendoyeritcylin- 
drMs zi dd cyUddres a/, gh>v/ln ddxxiyfiMe Mt ofa 4/ 
m d ; ^; MtfeUiMm dcMtMM» a u b ddf»U^m dcutMm c u d ,- 
fiMeMtcyUnirns al ddJiUdMmdeMtMm cud. AeqMdles erg$ 
fMMtdM9fimnlcyUairit\^^\\^dCMt6feUd» cud. Hemftiff. 
dqMdUhMSfMempeeyUndr» al &/eUdodc'Mtoz\3i.h,remdMet^~ 
Unirns g h ^qMdUsJrmfle c a b d . S^i&c- 

VerfMSMerticemmerotimitdtiene ofns efl . EB» HdtmnfeU- 
inm dCMtMmftffMmfldnec d , iehedtqifMmifrMSnm c a b d «<r 
fttsMerticem^dqndlecyUhirp^idte gh ( inmmoie cyUnirMs gh 
minerfit tyUniro c c d f . ) 

Hdt^MtcyUndrMs ed di^hyitdreSfdnfddtdy di fl,c?' 
«r^^4 1 b , iicefrHBum c a b d ^e^iir^^ <f^ cyUndre idte g ji . 

Ndmre0alii di n\f eflnt dn:idibo,fiMCMtdCMtMmfe' 
Uium cuddidcMtMm ZKihi&perconMerfimemrdtieMis u 
i di i\t erit nt 'deutMmfoUiMm c u d , ^mt^ «/ cylinirMs ed dd 
/ruJfum c a b d . J^/«/ nfdifi^itdefi etidm cylinirMS tdem 
c d <(</ g h , dqudntur ergefrufiMm c a b d ^ «/ ryUnirMs g h. . 
^ei&c, 

' ■ ■ . SchoUum . 

* Ex priori parte hnius dcmonftrationis patet folidum hyper- 
'bolicumverfusinfinitamflanitiem ^/'magnitudinc infinitum 
'^cfle.poteftenimexipfofumiparsipfius qux xqualis iitcuicit; 
quc lAtigaitudiai datfe : 

Coiol 



ProbleniaSeetindunt» ^ tx$ 

Ck)rolIa(Juia XIX. 
• 

XjltftUdimiuuiiimfiSimfU. 

m ab. Ofirtit illiidfenre ittri 

lUieftiuie pj> itsvtfiitfiiim api 

b ndfyUiidrHmfiticirciim/iriftBg 

fitvl c nd Ai dumuudc rdtii c | 

Mt A, Jitminmt imijiiMlitAtii . c' D 
riiit,vt c ad i, ittdM* ef 

^d f g ; &fer g dncdtifffUniim p 

hl. Eritfi c nd i, vt el dd{ 

%,iimfi(fh(qiidlidreClMifiiU)vt ig td be, hKeBvtfrn 
ami ai ddcjUmdrtm al. Sgoddrc. , 

Si veri ddtim fUmmfecMiJit pi, li- fiUdimfeciimlmm 

Jitiiifaiiitverfiis f iterumeddemlege ,itifncedtmiu . fiia 
■vt c td i, itt tttdddttm l^.&ftr e dtKdtur fUmim 
ab. Sritfifinaim ai ddcjUndnm al,i»» gi «< eb.jK- 
•<w cf *< fg, bicifivt c dd i. Sgidirdt&c. 

CoroBarium XX. 

SfilfiUdnm dcntimfilttmfUnt 
ab. ^flrtetillnd iterimfiecdrever- 
fu l.ttdvtfinBimiiiierfeaiiiut 
ciiiifrdkinfim,ddiiifcriftimfiticy. 
Undrnm qndmUiit ddtdm rdtimtm 
mdiiris indfUdUtdtis^^iAedt, vt c <\ 
dd i. 

ridt,vt c dd i, itdddtd eldd ■ f 

fgi ddffiaifer ^fUm ib. rrit . ^ 

finffnm ib dd cjUndrnm infcriftim ob, vl gb 4^ eb^ 
/wv/ ef dd fg, /«« c */ d. Sgtd&e. 

SiverifUnnmfecdntddttmfit, ih, &fecdmd»m fitftUdi 
u<rimiiidtmlegivirfiuinfiiUdmUiiiiiiidi»im.fidivt cdi 
i ' dit* 




d iu ti dddtttm fg. £ruq,fn(fitim ib ddcjUninim 
h vt ei M fg. "'TfVt « M i- atidd&c. 

Corollaita». XXL . 

Efiefhi!himJiciitif0lidi3Lhci^ v 

fotntttnrqicircHlnt.ci mediiitfr» 
fgrtiattdUi intrr hi^et ad>bc,^ 
criiMtir cylinirils eg ctliufctotqi 
Mttliiiiii . BictfrHftum ac ild 
cjUndrtm egefcvtrcat i\dd 

Fimemimt/trcSjt ad jd bc 
it* il iT./ lo, d- dJ Mtittidiiic 

lo erigiaiircyUmiriu in,qtiifi}»^ii, mffmftt ic {fcric- 
rtU.i). } Jdm cylmdrjit an W eyliiidrmi i%, rittiewem 
/MketcemfcfitMtexrMietiehit^tim^nemfe qutdrAti^lxA ndeii 
iKeSlexrttitiuKfit ad 4i\>C; fniifiitiian^f il *< lo 
&exrMaiieidtilttti$itlim,*lmfe !<, td'^l^.: f)r^cyli*Jtmi. 
an ^^ eg, eritvtreifd il rf^ fg,- profterea etitmfrxfii 
ac ndcyliiidriiiH eg <!•»»« ij */ fg'; Snpd&c 

Settlinmi^'' * . 

Ergo fialtitado J^ fiataiqaalisiiiift >/ «litcylindnK 'f 
xquaJisfiiiilo rfc, -'< . ' 

. Corollatium XXIP. . . ' 



''£IhfruStlmMiitifoUdk^t,\^6^^'^ 
4 , qmdbahejtdlterAmexfitifi^ .ii 
fSAafijii£iii)i^iUafh)fiit4 .jdi^-^ '■ 
fqiialem bafi e idt^ndsi t^.Si ' 
ctfiiftum ac tdcyUitdmm c%ef ■ 
ftvtredf^iillimfiA ditttietrt inp ., 




Profalema Secundam # i J i 

q$idlis bdjis^ drfnb dtitudine frmHi ^ ddrecfangklMf fier Mxem 
eylindri. Nemft^treBdngmlmH bc, hi jdreffdngttl^tm cg. 
Fidtvt ad dd bc itd hi dd io;ere£foq-,cylimtro al 
cnm dltitndine io, eritfrttfNtn ac ftjthdeeylindre zX.Iim 
cylindrus al ^/ eg, ^^ stjmdles bdfes^eflnst oi 4^ gm, 
Sedratio rectd o i 4« g m , comfonitftr iM rdttone reifm oi dd 
ihj fiue hc dd ad,^^r ^ bc itt/ em; c^ ex rdtione h 
i ^^ gm. Ergordtio oi 4»^ ^merittddemqudeftre^dng. 
b c , h i ddreddngubtmfub e m , m g • Frtftered etidm cylin 
drus al, ftuefrtthumzc dd^cyUndrum eg eritwre^f^ 
gulum bc, hi, ddreUdngulum em^. £iuod&c. 

CoroUarium XXIII, 



B L C . — -^ 



* 



A V i>E 




Sifrulium dcutifoUdi a b c d 
&cylindrus t i dqudtes dkitu- 
dines hdbuerint . EritfruHum 
ac ddcyUndrum cfwreifdn- 
gulumfub bc, ad, dd quddrd^ 
tum eg. 

/'w/1// ad i<rf bc, //^ lu 
4^ u o . eritq;fruftum a c 4 f/Vii* 
lecytindro ai cuius dltitudo fit 
u o . /i^w cyUndrus a i iiJ cyUndrum e f , rdtionem hdbet co^ 
fofitdm\ex rdtione dltitudinum uo dd ^f; fiue uo ii/ ul 
fiue hc dd zd;nempeexrdticnereffdng. bc, ad, ddqus^ 
drdtum a d « f / ^at r^//tf;!rf ^^/iifXKi ^ /nm^^ quddrdti' zA dd 
eg. frg^^ cyUndrus ai, fiuefruftum ac, ddcYlkuktmo c<i 
eriiutre£tdng.fub bc, ad, ddquddrdtum cg. ^od&c. 

CoroUarium XXIV- 

Fruftum dcutifoUdi ihcAj dd eyUmkttm qumtihet e f , 
rdtionemhdbetcomfofitdmexfdtionere£ld»^tdi bc, li Wr^- 
&dngutum a d , g d&exrdtione quddr. ^ddd fudirdtu e g. 



X j X Oe Solido Hyperbolico 

' FUt vt ad 4d bct iid li 

qudUs fruHozc. Jdmre^M io 

ddTiHdm g f , ry^ vtt4£ldHgmlmm 

fub bc, li dd ft&dHgulum fuh 

ad) gf« tndmTdtiorcHd \o dd 

g f ) comftinitMT tx TMU(ue io dd 

i\y Jiut bcddadsS^txfdtieut 

il dd gf. ETgoTtiid io4^gf> 

til^tTt^dngulum hCyiX ddTt-^ 

Bdngulum ad^ gf.^ SfdtyUndruj au ddcylindrum c£ rd- 

tionemhdbetcompofitdmexTdiiene io ^^ gf» nemfeexTd- 

tioMTtfidnguli b<:,.li, ddreSidngulum ^A^ ^U&exTd-^ 

tione quddrdti 3,dddcg. pTOpteTCd etidmfruftum a c ddcj- 

hndrum e f Tdtionem hdbebit comfofitdm ex Tdtione reBdMguU 

bc> li> ddreSfdngulum adygfi&exrMionequddrdti adjb/ 

cg* ^od&c^ 



BL C 

/o \v 


/ 


\ 



A 1 



I>-B 




Scholium^ 

Poterat etiam proponi fic . Friiftum ^ r ad cylinclrum r/, 
Tationem habet compofitam eKrauone re<5tanguli ddy ily ad 
rei^ngulumi^^/l^; & ex ratione quadrati ir> ad r^. 

Corollarium XXV. 



Simtduofruftd dcMOTumfoli- 
dntm^^MdUdcunqm * Dicofru* 
.Hum^hcitJidfruftum dfga» 
hdb^re TMsontm €ompofttdm ex 
rMi^me Tr£fdngulorum i^4^um% 
^ ixrdtione dltitudinum\ nem^ 

• 

rft exrdtiomtre&ditguU b c^ h e 
-ddreffkf^jdum fgy da;^f4f 
wdtionert&^ a«i dd .mi« 




U li £D L A 



Itdt 



Problema Secnndum - 135 

FiiU enimfiifer bdfiA^ cylintlrus do cmm dtltititdine 3^0^ 
^fMitJtt ^e^MAlis ipfi ni . Xritf, (ftt C$r9U.2^.)Jrufifimhcdd 
^Undrmm d o > f// re^dngulMm b c * h e Md qMA<ir4tMm d a . 
CylindrjtsdMtem do4d/rMffMm d^ efi {perCwU.22.)vtre-. 
aangtUMm ^ a o ddreHafigMlMm f g, m I . N^mfjt MdilUidt rd' 
tionem hdketMemfofitdm ex rdtione rt£i£ d 3.jU f g , fiMeexrd 
tione qMAdrdii d a Mdre^dngMlum d a , fg . Et ex rdtitne re- 
tTrf osLddmlj fiMe in dd m 1 . Jtatia itaqifrufii h e jidfrM-. 
Jlum d^£i>mfonitMrexrdtionii>us,rc^aii^MU hc^ he,ddfMd 
dratMm da; t^ exrdtione^uddrnti d 3i MdreifAngulMm da,fg: 
f^ ex rdtime reHd ixi ddm\. Hemftoq. medio iUe termittofii- 
ferflMontmfeqMddratod^. Eritrdtiofrustihc ddfrurtum d 
g c9n7fofitaexratienere6fdngMli\>c,h,c, ddreSiangMhim d 
a , i%i&ex rdtione re&a inAdmL Sspd erdt &c. 

CoroUarium XXVL 

EftofrMftMmfelidi dcstti a b c <1 feM^ 
flano h 1 ; dMCdtmrfib^fdrdlUld dd/txe. ^ 

Dico , totiimfnifiMm , a b c d <Mf fdrtem 

hhciycjevt d.add\\i. 

NamfoUdumdcvtum z^d tdifoUiu 

bgc, eftvt 3.i'adhe,fiMer/t ii ad f 

n ; (^ diuidendofruftum a b c d ddfoii- 

dum acuium h^c^rtfvt znddnf,fi- 

me vt an W io . Soiidum vero bgc ad 
fruftum hc (ftmtUargumento) eftvt oi,adi]\.£rgo exdfuff, 
/ruftamzc,dahC£ritvt3LQddhi. ^»d<^c. 

Scholium . 

Hinc jKitet^otTtodo dattim iruflnmacuu folidi in data ra- 
tione lecari poiRr,(]Uod tamen ad finem Corullariorum elegan 
tiori problemate cxequeoiur . 




1 54 Dc Solido HypafcdRco 



Corolfarium XXVH, 

BfipfrMftnmfdlidi Maii abc d . ^ains 
dxis m i . fttf.eemirMmhyperhtsfmnSfM 
h • StctturdcindefmftHin a c flan^ qup^ 
i;$inq\ e f adaxem creifg . BicofrHSum a 
f , ddfruftnm tc.effevt rf£fdngtilnmf$tb 
il, hm, 4k^r^<f5Fi^/r^W/!r«iy3iri hi, Im. 

NdmfrnHnm zi ddfruftnm ec,r^/i#* 
jiim )&i^^^/ comfofttdm exrdtio^nefrufti a f 
dddcueumfoUdum egf; & exrdtiBneftUdiUngi egf, dd/irm 
ftum e c . Sediftndfolidufndcutum SL^d dd dcutum fplidmm e 
^feftvtrecfa ai 4<^ cl;//!r^v/re^ii Ih ddhi^eritdimide»* 
dofruftum ziddfolidum egfvtliddih.AmplimszSolidMm 
cgfddjolidum hgc eJf vt el ddhm,ftuevt mh ddhU 
(^perconuerftonemrdtionis, eritfolidum e g f ddfrmBum c c, 
1// h m i^^ m 1 • P4/^/ ergoquod rdtiofrufti a f ddfruHmm e c , 
€omponitmrexrdtione lidd ih^ c^exrdtionehm ddpil Pr^f- 
teredfrulimm ^idd ec, m/v/ reitdngmUtmfuh li^ hm, ^^ 
teifdngulumfub i h , 1 m . ^od(^c^ 

Schpbttm m 

Ideo fi £at, vt ;^^ ad /(/, ita ^/, ad //• Bifariam fecabi* 
turfruftum dci planoperpundiim / du<3:o# Aequalia enim 
erout ipfa re^langula » 



Corollarium X X V I IL 

Si dxisfruHi a b c d bifdridmfe^ 
€etur}pldn0 e f . Eruntporttonesin^ 
terfeynemfc z,£. ddtc a/treifd ad 
dd hc .fcilicet vt didmetri bdftmm 
remotdrum » 



\-^ 


S c 


-h 


V 


/c 


\ 


A 


l 

1 



/"r*- 



Problema Secundam 155 

Trftfit/m emm a.i sd cc^ tStvt reCtdngulHmfMb h g» o i <*i/ 
re£iMngulmmfith\kOy '\ ^ ftr ff£ced: Sed'oi,^i^, Mltitudu 
nes ri^fangMhtHmfknt tqudles^ ^rgefruBum a.idd cCt erit 
•vt gh adliOyfMevt ad ad hc.^oddrc 

ScheliMm^ 

Hinc pate^ quod fi in folido longo hy 
perbolicDquotcunq;fumantur asispor- 
cionesdeincepsarquales 4yh,e,d,e. vbi 
cunqjfiatinitium. EritfruftMm^ ad^g' 
i6 vtrcda/4 ad -^f . Frufti]lm ver6 gh 
Ad fii ericutffij ad /^t^. &fhiftum hi 
ad il uihsy adyr.&ficininfiniium. 




CoroUarium XXIX. 

Datum Aeuti fotidifruftum 
a b c d /« ddtdratiene fec4re; 
futdvt ead{. 

Fiat , vt reSd a.d adhc, 
itd e adalidmqudfit g. He- 
indefat,vt zad i,it^]\i4d £ G FA 
i i> &fer i dueaturtianu m n 

lamfrufiam ntidd mcefi 
•vtreifangultfm \'Oy ih^ddreffdngMlMm \i,oh. Ergirdtiefrt^^ 
sfi Sinad mc cemfemtur exratione Uttrumlo dd oh,fiue a 
d ddhcfiuec adg. £t exratione laterum hi ddW^fiue g 
adi.Ergoratio/rufti zn ddmc, cemfonitur ex ratiene e ad 
%i&%^d i.Prepteredtrit znfruftum dd mc vt e dd i. 
^d&c. ■ ■ ■ 

limifta fufficiat demonftrauifle, ex plurintis. TheQKmati-- 
busjQUxex^undiilimo hocfolido deriu^iripotcrant. Inte- 
rimadpromiflamdemonftrationem accedamus, quanitamen 
pricterire poterit quicunqi iainaUati coDtcneiu iiierit . 



«5« 

2>f DimenfoneAeutlfoUiiHyperMick 
iuxfamefhQdum ^ntiquorum^ 

SVpereft niinc vt Theorema illud> quod poft lemma QoiiD. 
tumoftendimu^ permethodunij&dodrinam Indiuifi- 
bilium , demonftremus iterillm more Antiquorum , & prascipuc 
Archimedfs « Impoflibilc enim quodammodo videtur » inHoi- 
tamlonginidine nguraoifub folita figurarum inrcHptione,& 
circumfcriptione poflfe comproehendi. Tamen id non folum a 
nobis fadtum cft, verum etiam a Clarifltmo viro > & Geometra 
prasftantifldmo Roberuallio^qur noftrum folidum hyperbolicii 
inuetis arduis» fublimibUs, acuti(fimis,& vt breuiter dicam fuis , 
menfurauit, eiufq; fruftum in data ratione difTecuit. Abftineo 
ab iliius demoftrationis editioneinuitus • comparuitenim eius 
epiftola eo prorfus tempore > quo iam h^c pr^ lis fubijcerentur, 
neque de voluntate Authoris (atis conftabat>neque iamper te- 
pus Hcebat expe<^re, donec illius beneplacitum exX^allia Pa« 
rifijq>fignificaremr. Veniamus itaq; ad lemmataopportunai 
quorom primum fit . 

LemmaPrimum* 
Diffttentid^ qud eft intet iuos circuUs , dichrttdttm qmemli. 
bettertittmieJtvtrectdnguLitm comfrdhenfttm/ttb JiJJirentiM^ 
& dgffregdtofemiiidmetrorttm eornni^m circnhrmtt^qMdird- 
tnmfemiiidmetri tertq iUins circuls . 

yoceti^ dntem tdlis iifftrentid ittornm c^tnlofmn^ qudnio 
* €oncetttrtcifnerint , Armilld . 

Efto Armilldjine Hfferentid ino^ 
TMmcircnlornm concentricornmyil^ 
Idcninstdtitnio ^hjCentrnmverh 
C . Z>ico Armilldm a b , dicircnln 
qnemlibet ddejfevtreiidngnlnm 

abe^ 




Problema Secundum 



ixp 



abCt ddqMddtdimnftmidiMmetn df* 

Ndmcirc$tlMS€xr4di0 ac, dd cprcMlum ex rddi^ ch, ejf ^J^^ 
WqHddrdt$m$ ac, ddquddrdtmn ch\ & diuidendc ArmiUd ^^l/^^' 
ab, ddcirculumexrddh cb; eri/Mre^dugulum abe» ^i^ 9»uM 

cb, dd circutum ex ^^udi. 





AQEYO FiLH 



qmddrdtmu ch. Cireulusverbexi 
'rddie df» efi vt quddrdtum ch ^ddfUddrdPumAi. Ergpex 
Mquh i erit /irmiUd ab, ddcirculum df» vtreifduguldm z.h 
e> ddquddrdtum df. £iuod erdt (^c. 

Lemtna IL 

SiexeyliudrcreSfe ab, ^i^* 
idtus fucrit cylihdrus cd» ^/r« 
cdcommunem dxem ic conHi^ 
tuius\reliquumfolidum excdud-^ 
tum quodremdnetydqudte erit cy 
lindro cuiddm reSfo fg, cuius 
quidembdfis fh dqudUsfitAr^ 
milld » ^4rf rirr^ centrum e iW/#- 
tudinem hoBet a c > dltitttdo vero 1 m uqud&sfit dltitudini e i. 

Vocetufdutem tdlefolidum excdudtum^ tubus cjlindricns . 

^oniamtres cylindri a b > c d > f g t dquedltifunt ; JSrit 
cylindrus ab ^tfi^ cd, vtctrcnlns ao ddcirculum cu» c^ 
diuidendo erit tubus cylindricus dd cyUndrum cd, vtdrmiUd 
ac ddcirculum cu ; fedcyUndrus cd ddcyUndrum fg^eJf 
vtcircuUis cu ddctrcuUtm fh. JErgo ex dquo erit tubus cy -» 
Undricus ab» ddcyUndrum fg, vt drmiUd ac dd circtOm, 
Ufm fh. SeddrmilU ac ir/rr/ifi!^ ih fupponiturdaudUs;er^ 
^d^ttAuscyUndricus ab» fqudUseritcyUndro tg. ^fMf 
erst&c. 

Lemma II L 

SuiUbet cyUndrus reCfus ab, dd quemUbet tubum cyUn^ 
dncumreifum cd^ rMionemhdbetcojnfofitdm exrdtione dl^ 

S titndi^ 



V 







AH 



156 Defi&UdoHypeHiGditeo 

titudinHm j nemfe ch dJ f d) « ' 

. d^ vx Tdihni iajkm i nmfr 

exratidnejdMlt^Mil Jih Mdre-- 
tik i : ^dngn/Mfh c i f . (^emefiftrjt^ 
tktn enim fft itd (tffe eirttUtm 
a e addrmittam ci^irt qtta^ 
dtAtHm ah ad rettdngHlHm 
cifO 

PifadtHreytindrus ImjCH- 
tHs dltitnde nmfit AqHalis dltitHdml f d > bdjis ^era 1 n » sqH4 
lisjit drmilU ciy £t erit^ perprdcedens lemmdy tmbms cylin^ 
dric Hs cdi dqndlis cylindro 1 m ^ 

Idm cylindrHs a b , ddtHbnm cd kdndeni kdbrbk TMft^Me^ 
qHdm hdbet ddcylindrnm 1 m ; t^i^e van^afitHm ex rathne dl- 
titndinis cb dd nm jjlne dd iA\dr exrdtione hafimm >,^hec efi 
circHli a e adcircHlHm 1 n \ fiHeyfHddrdii a h ddfHttdtktwm 1 o> 
VclqHddrd ti a h , ddreifdn^Hlwm c i f . ^iifitdtrm J?r; 

Lemma IV. 
p;.-.^ kno hypnbdd cHtHs ^yp^pmifiiH 

ab, bt\»»gHlH7h reffkm cmfrfhi^^ 

denies ;fitqHe hyperb$U fcMdxi^s b d , 

(fe^iaicM dffem , qnia b pknifi»f9^ in 

Wo "dfsi^ftati concHrrmt , Cdni^i^- 

yerbolh eti . ) Inch qkddr)t(Hm Yeti^ 5 ^j 

"jH^plHm epcUiufifHhlfkrreai^guii^^y 

%terafy1nptotos , c^ hyperbotkk J^jf^t^imj^i^enfii 

DVcantHr d c , d i kfyUptM^^^iMjime^ ,• ^i*f • /^^ *> i 

^Cs^quTidfdtHm: cit)hahg}iti^ad bfelhii^i^^iftdifdcy^ 

dd i r^^/ , /^^^ quddrdtHm line^ b d , duflHm erit attddrdii 
r^V *' ^ ^ ^ ^ ^^^^ reBdnguli a e , //^/^ dfymftotos , ^ hyferMdm if^ 

Lerrtfha V. 
EHohjperbold ab , cHt:isdfymftoti^g^Hmre&Hm eonti-^ 

nen^ 




B E C 





dMhusfmmSis a » b , vtcMmqi 
imhypnboU » dmnuuw du4t r/* 

iiuht^z\t^fmfm$ cd^ <*- 
quidiftantts , <^ a n , b m dltt^ 

ri dfymftot0 d e fdrdllelf^ qUd 
concurrdutin 1 ^ Tjrmi ctff^cr-' 
tdtmvniuctfdfigUTdcifCd dx^ 
cd# 

Dw cyUndrum quemddm i 
epo (cuiusquidcm bdfis io 
hdbcdtfcmididmetrum i t ^ j^«4 

lcmfemidxihyperboU\dltitudoverp fit intofoeftd ie .J ^4^^- 
remefietuboillocyUndrico^ quifitexconuerfioue re^4nguU i 
b r/rrii ^x^/» c d ; Minorem vero tubo illo quifit i?t fOifltfrfiottf 
reifdnguU i 1 , cfrcd eundem d^em reupltfti . 

Infrimisiquid it eSdqudUsfemidxi hyferbpU % ifff ^^^r- Lemffg 
drdtum i t dufUim reSfdnguU d b yfiue dqudle re04t$g^lf U b • ^^^ • 
lam : cyUfulrus o e , ddtubumquifit exreifdtfguio i b (.^/^A 
Ugefemfercircddxem c A)rdtipnem hdbet comfofit^m fx rd- ^ il 
tione bdfittm ; nemfeex rdtione quddrdti i t ,fiue rpSfj0gufi U h% 
ddreffdngu/um u i e . ffoc efi ( dbieffis reffdttguUs) exrdtf^^ 
ne Uteris ue dd ti\ & exrdtione Uteris eb d4 iu.. ff itrfit^ 
fer « rdtione dltitudinttm ; nemfe re£f( tiddth. Btgo r4tio 
fjUndri o e ^dfukttm i b , comfotfitur ^ ptddi^fs mkfH ^^- 
tiptfibus.fciifftt. exrdtionere&d uc 4Jci: & e^/jfffpgf ei 
dd eb;^ exrdtfone eh ddiu. froftered cyUndrtfs ffCfdfl 
^ fffb$m i b f^it^fffrimusterminusddvUimum; ue^fc vt re^ 
^diicddiiiihfc eft minor » ^goderdt oftetfdffidttno ftimk . 

M4M verh tylin/iri o c , ddtiihm>iqt4ff f^J049gHlo i I , 

^f^m!^ y^> ^^^4»&tkfm fti ^vk^efi '(aa(mf*,efi4Mg/t' 
U4>iit»f^.nej,^isii , V ic; & pii^rAfifipene^qtii ((ttjeris a i, 

.44iii^.:^i>ifii/er^eicr4ififiMe4iiff(4iftfmh 44 ai • Mr-, 

S 2 g»rd. 



tit 



1 ji, De Solido HypJerl 

£9rdii0 cylindrioe, MdttAnmilyC^mfoMitiirexbistf^MS/rSr 
JiaiffdtiottikMtinemfeexraiitmeiiiidi^y&tfi-^d au xf 
zidd.iw. Frmerurjlitubrus oe, ddtidmm il, mt^vtfru 
mttftermiittts htidvltimMmiu. & ide» tniiur , S^d er^ 
§ft£»tb»dttm&e. 



Letnmia VL 



^ \ 




DDCz; 




% 



OjSTV" 



1 

£fio hjferhoU culus Mfympto- 
ii c d 9 d e Angulfim tfifum com^ 
frdtheiuldntyfMmftifq. in h^fer^ 
hoU^tcumqnedmobns funiHs a 
^ b'yi/ucdntur ai, htJ^mfto^ 
to cdfuraMcU. 

JbicoJoMum illud dnnuUrt 
quod defcriiitur tx conuerfiont 
quadriUnti mixti [iabe, cirtA 
M^tm cd rtuolufiydqudttjftcui 
ddmxyUndrottBo i <• po . Dthtt 
duttmhuius cylindri altitudo tjft 
\ e ; didmettrvero hafr i o , dtbtt 
tffi Mqudtis int^4f Axi iffius hy- 

ftrboU. 

Sit inim (fipojfihHt tft) folidum tUud dnnutdrt fdSfum tx 
* quddrHinto i a*b e ^^dfta dxtm c d rtuoUtOy minus xjlindro o 
r ? (^rfendtur dtfc^ius aqudlis cuiddmfoiido K^ 

Stcttur bl bifaridmin f . deindtreliqud {\ftcttur hifdrii 

in ^^Eth^cfdtftmftrdonectubuydliquiscylindrictts^ qmde^ 

fcrtbitur exrtuoUtione rtBdnguli alg,, minor fit foUio K. 

'^umemmftH^tota hlMfdrttraqnaitsvltimf gl, ducan' 

fur ^ finguVtsfunRis diuifionum ^ rtSta g h , f n jyi"^ dquidi* 

fidnttsiffi de. £x funSii^^n^trh m, n r- w quibus prudiBf 

fdraUtifhyftri^oUmftcanty iemitfdnturreBd^fiutfotiusfU 

Md miynifXiX^ddd/ymftoton dxerjtffd. Dcniqut ex <on^ 

-■ ' ^ eterfio^ 



Pi^obleiiia Secundum « i jj 

Mcrjione finguloTHm rtifdHgulormm ffudlinmj quwum vnumtfi 
a g ytotid€mtub% cytindmi dtfcrituntur circadxem c d ^ 

Jiun: Jtubus quifif ireildngulo xh ( intellig^ femper circA 
4tHem c d^ ob dqudlem dltitudinem , euttdemque bafim y aqua/is 
erit tubo r f • Jtdditoq; conmunitubo r n . erunt dtto tubi b r>r n 
fimulfumftiJtquMestubo ny^Jiue tttbo ng.Addit9q.commUr 
ni nm. erunttrestubi b r n m , aquales tttbomf^fiue m 1 ; e^ 
Addito communi iHtimo m a , rriv^/ onmes tubifimul b r n ma , 
fquAlts tubo a g , nemfe minores folido K • obconSruifionem . 
PrMptered vniuerfdfigtir/tfoUdd coniians ex tubie er , & n , z m 
X a > ciHumfcriptdfolido annularifa£lo i quadrilineo i a ib e, mi» 
nus AdditfupraipfumfeUdum annulare^qMamfitfoUdum K.£r. 
go ipfdfigUTA circumfcripta adhucminor erit cyUndrars} c.^od 
afi abfurdum . Ham tubus a x • fuperat cylindrum xo; Tubus lam. tl 
item m Tfuparat cylindrum zi.drfic dereliquis pn lemmas. 

P4>ndtMrdeinde(fipof^le efi)foiicCum annularegenitum ete 
quadriUneo i a b e , matustjfe cyUndro o e .ponaturq. excefius 
aquaUsJpadoiCuidam K • 

PiragaturfimiUs conSruHia , vtfttpra\ ita vt onmes tubi cyr- 

Undrici ht^^^j^it^eresite^um^BattdanturfoUdo Jc.. Xttn/c 

^enimfi^rkiffcriptainfoUdoannuleiripra^iQo , confians ex^tft 

i/> & b , z r , X n ^ i m , minus deficiet ab tpfofoUAo annuiari^qujt 

fitfoUdum k • Proftereatddeminfcriptafigura adhttc maior erit 

iCylind. o 6.^od cif abfurdum.Namtubtts]yi\i ndnoraficylin- u^m^ 
^dro X o \,&Jtbus x n minor oficyUndro x t ; Etfia dert Uqmis . 
Pataergo^ , qtibd foUdum Aimulare gemtuhtf^ co^tttnfi.^Me 
quadriUttei i a b e , circa axem cA , aqualeefi cyUmkos) t> J/- 
quidtmoHenfum efi^ neqi minus , neq. maius effepojfe^ 

* Lemma VIL 

Ifiohypnbold^ ^uius ajymptoti angutumre&Jtm.cmtdnen^ 
tesfiniz, b , b c ,* &^onuertdturfiguracircdaxem ih^itd utfi- 
etfiUdut9t^kypefboUctnniCuitts.infi0iufitlangiiudo ^wtfus^ar^ 
tttx* SeffodeindeAuiu/modiJolidOyplatto de ad dxem ete- 

cioy 




1 54 I>e Solido HypcrboUco 

gc, habins dmtuiiutm dS, IntMg**»' 
ametUustyUtubmt b g \\ifiiius4titH4^fit 
b g , bafitverhfemidiMUter b O feiutMr 4" 
quaUsfemidxih^ferbeif , Di4& (yUudrum 
b 1 duflum ejfeiyUudri U* 

Ndmefiiftdrus b I Adtylindrum fe,.M- 
/M0e» ^<^/ cemfcJitAm ex rAtiw* k4^. 
eittm i uev^ tx rdtiotie quddrdti oh M 
h%i&ex rdtione dUitudiuufi^ , Uemf* (PC 
-Tdtitueredf b g -«a^ g c , siueptdtksti bg 
ddreBdugultm b g e . Ergscyliuit.Uf ^ i * 

4d<yliitdrum fe, efivtqudd3r4mt'&^*dd'f^*»g'^'*^^' 
HfnfedufUis. Sliuderat^c,.., . 

Efto hyperbola,cuius afy mpto- 
^ Engnluni reda contkieQites iim 
^iydc^ £t fumpco iQh.Xperbo- 
la<]UolibecpuniS:D dA^s^mx d 
^ parallela ad b d. Timi couerr 
tfttor figura cif ca a^cein jUf\ix3L 
Mt fiat folidum acuura byperbo^ 
iicum infiiiitqe longitudmis ver* 
ibs partes ^, (intellige feoafier 
pun£tum h ininfinitamdiftantiE 
cflTe remotum. ) Conftabitq. pi»' ^ 

di Aum folidum hy perbolicum ex duobus foIidis,nempe ex cy- 
lindrore<^o/^^r,&exfolidoacuto ebd^ cuiusquidem ba- 
fis erit circulus edy altitodo ver6 fine finc . 

Dico vQiu£rfutn.huiufnaQdi-&]fIidum/^^^^ cy- 

lindrOiCuidamredto 4rif^«cuii2S.tIiitttdf>%^^Y<)^^.^^u- 
diameter bafis acuti folidi) diameter «tr^ baifis ^...«^115 fit 
integro axi hyperbol» . 

Sit 




Problema Secundum . -1^5 

Sit emm fCtpDftA)ik ^d) folidikn hyperbobcum/<r^i/( ini- 
nus cylindro 4i . Pon^uorqi cx cylindro di cylindrus alicjuis 
ncil, quiaMjualis fit Iblido hypetboblico .• & producatur Inm 
doncchyperbolae occurrat in m. (occurretenim,cuin aiyiD- 
ptoto db fupponaturparallela.) 

lam cylindrus H %r iequaUs o-it rolido aimulxiiquod defai* 
biturdreuolutione^uadfilitieimjxti i$mdt\ &propterea nu-^*-*' 
nus otnnin6 erit folido itircgro hypcrbolico/<f tdc. Non er- 
go cidem eft xqualis . Quod eft contra fuppolitum . 

Ponaturdeinde(fipoflibile eft^ folidum hypcrbolicum/« 
bdc maiuscylindro rfi, Ql<«liam igitur folidum hyperbo- 
licum/ifJ^i^f •(fiuefinitxmagnitudimsiit, (iucinfinitfjniaius 
fupponiturqudmcylindtus .«i. Eritaliquod ipfius fegmenti], 
puta /<*'»«<■, aequaletyhndro di. Q,uodcftabfurdu[ii. Na" 
ioiidijniantuilanefiii^in^euokinonequadrilinei mm i/r» f- ^•f: 
qualeeftcylindro ;!»»;CyIiadrusautem ^ » fubduplus cft cy- 
lindri n h . Erg6 lota portio folidi hyperbolici/f omdc^ mi- ^■*'- 
noreritcylindro di. '. 

Pater crgo, qu6d vniucrfum folidum acutuni hyperbolicum 
febde^ quamquaminiriitsidngitudinislit, «qualcramcnFft 
prad jdo cy linoro d i .^Joandoquidem neque minuf» neq; na- 
ius cficpoteft.QuodctMoftendendum&c. 



APPEM- 



?5* 



• • 



ATPENDIX 

7^e Dinunfani CachU^ . 



CV M adhuc i nemine, quod ego fciainyGeofxietrica con- 
(ideradoneexamiaatum fic iolidum vulganim> & and- 
quilfimum , meooi iudicio aliqua animaduerfione non indigpii 
( Cochleam imelligo , ) noh aos re fore iudicaui iUud brcui co- 
templationeprofequi. Non enim aliena erit apraecedemi li- 
bello praefens fpeculatio , qua? per Indiuifibilia cunia»fuper& 
ciefqi'cylindricasprocedit« Neq; ir^ratum Gcometr is opus 
futurum exHlimOy fi demonftrauerocui figura? notas iam dimen 
fionis» aequafe ficfolidum quiddam neque redam, neque rotun- 
dum^fed fpiralireuolutione concortum,qualenullum adhuc in- 
ter menfuratas figur^s poflidet Geometria* Pr^mifla itaq; defi- 
nitione veniamus ad rcmmata, qua fieri poterit breuitat^espe- 
dienda « 

Definitwl 

Sl eodem cempore moueanturduae 
plan2figuras,qua?fen^er ineode 
plano confiftant, nempe redagulum mS 
€d. circa axem/^ i mom circulari a?qua« 
bilif & figuraquaecunq; de motu pro« 
creffiup fuper latere Jc . Solidum quod 
afigura genitrice d / defcribitur , Cochleam appello • 

LcmriuPrimum. ^— 



SBofiUdtm qMdiibet rotMndmm a c b g ; r /r- 
ius aicissit a b, fgwdgimttix a b c \fe£Hq . /// ^^ 
fUna dfc dqmdifldnterdYii^ dr dd figttrdm ge^ 
nitricem ereffa , qnod qmdemfdcidt infnferficie 
plidiTotnn^femi/eitienemlinedmdit* Dic$ 





f^ 



. DeCochlea 14;^ 

€ircddxem <le> dqudrif$Hd$qM$ddefcribitfir k fgwd Act 
€irc4dxem ab reu$lutd^ 

Inttlligdtur enimf$lidum retttndumfecdri dli$ fldn^^ftr c £ 
^du(l$ycrdddxem Abereff^^ eruntf^funffd cfg infemi* 
circuli feriphdrid cuiur didmeter ejf co; (^ ideh quddrd$un$ 
if dquale erit reStdnguW c i g , &fr$pered (fer lemmd frimm 
frfcedentis dem$nflrdti$nis ) circutus cuiusrddius if, dqud* 
lis drmillf qudm reSid c i defcribit circd axem a b • JE/ hecfam 
^er verum erit vbicunqifitfUnumfecdns c f g • Brg$ $mnesJU 
mul clrculi » nemfefolidum r$tundumfA£tum i reuoYuti$ne jigu^ 
nf dfc circddxem d e, uqudles erunt omnibus drmittisfimul 
fumftis } hoc eHfolidofdfto dfigurd d c e > reuelutd circd dXfum 
ab.. ^$derdt(^c^ 

Lemma IL. 



B 

XI 






C 





£fi$cytindrusreffuxzbcdj dr « 
teBd e A tdmqudmtermino duareitu 
tineu infufirficic cylindricd aquales 
iffi ed moiiedutur : qudrum dlterd 
furo circutari motu Zonam e f a d ^^ 
fcribat , attcra vero quacunque motu: 
^$nam e h g o d defignans^moueatur 
d$nec amba advnumyidemque tatus 
cylindriyfuta a b feruentrint^Hice huiufm$diz$nds;fiuetQ^ 
narumfortienes iuterft ejfe fquates.. 

Concifiaturenimtrigonus cjlindricusfiiferior bfe tranfr 

firriydrfitfraii^eriorem gad cdlocari^itavtferifhdrid fc 

iffi a d /uferfonatUTf qua necefiariocongruent i cumfint areusf 

aqudtium circul$rum & reClffiue ch$rdd, f e, tkd (fi ducantur); 

dqualesfintferPr$f$fiti$nem j 3 .Primi elementorumEuctidis;. 

Iffd etidm ttSd fli congruet cum reildfibi fqudu a g , dllas 
iud reddfe interfecdrent infuferficieicylindrica, quod cffi n$ni 
ptefi . Iffdtdndemcurud h n e> qudtfcu»q^fit^$ugruet^umi 

X cuapuMi 




r^^4 g^<J -^ Kifienim congrMdli cBo : Effif g VdA tf^stMt^ 
€9fttt4k hn c , quA non congtuit cnmg o d ^duifJrq. i n infmper^ 
fcic cylindri , trit m i inf(jnalis ipfi i<Xy crgo etidm n 1 , 
qualiffit miyeritinaqualisiffi io ^quodefienonpotefii 
tnimferfuffofitioncm fqualcsfint i 1 » o n ^ ddditaq\fiuc ^Ia^ 
tacommuni lo, erittota io ^squalistoti ViX^ProftcrcatQtmm 
tfiangulum cylindricum h fe ffualeelitriangub cylindricu g 
ad .dridco ^fcrfrofirafhfrcfim^^na ci^d^Mnf chgd rjf 
ffualis. ^odf^c. 

Lemma III. 
SircBangulum a b > ^figura quacum- 
quegenitrix b c d moMcantur^^vit in dc^ 
finitione fofitum efidontc firaila inte^ 
gra reuoiutionc adidcmflanum redeant 
vttdc cefcrant moucri . DicofaStam co^ 
chleafrimarcuolutionis d g h , aqualent 
tjfc annuto circulariy qui ah eadcmfigura 
gcnitricedcfcriheturcircaaxem ae. 

Concifiaturcnimfigura b cd defcri^ 
hcrcfrimum cochlcam frima reuottitio^ 
ir/x d g h , quf initium habcat ifigura b 
c d , &fin€m infigura 1 f h .. Dcindc intclligatiurdefcrihere in 
nuUtmcircularem inferedeuntcmy quthattu initium^&fi- 
nem infiguracadem bcA^ 

Accffiaturinfigura b cd quaUbitrclia \o farallela axi t 
C , qufquidcmreSfa^ i o in reuolutionc duas zmas cylindricas^ 
(^ £qua/rs{ferlemmafrfcedcns) dcfcribetj invna^eadcmji cj- 
UndrtcaJuferficiCy alteram quidcmincochtea, attcra^ vcnin 
dnnuU • Btfqualesfemfer erunt^ vbicunq^fumatur rct^d i o . 
€rga omncrfimul^nfcjfiindricf quffuM incochtea^ fqndes 
crunt amnibusfimul \onis cflindricis quf funt imannftU , fro^ 
ftcrca & iffa cochlca fquaUs erit iffiannul^ . ^jtcd&c^ 

Corollarium . 

I 

HisurnMiufcftumeftoauicscocbleasprimas reuoludonis e£- 

fein- 




£ B 0]> 



DeCpcIilea« 147 

§e itiKt fe «quales ^^uandoquklem Gngdi^ ^dim ailaulo cic^ 
culari «quales fuurt 

Lemma IV« 

MJinintHtMs f/s qitd AfMamusfiiff^^ 
nhin X ly X 1 1 /& X 1 1 1 ^ ^imiC$^ 
nic^rmm^ EHoc$$ms a b c yftSlHs pl4n$ 
nonvnticdliper inVyfaciente in/ufer^ 
fcie conifeUionem f n r , qudecnnqy illd 
\fit\cutusdidmetertSl$ fe« JDttC4tttrq\ 
fi dequidifiansiffi^c. TumfiatyVt i 
t ttd tz (pdrtem bdfis tridngulifer dxc 
Hvtrtice c$ni auuerfkm) itd if dd fl. 
Dic$ f\ effeldtus re&umfeiiionis . 

Pondtur f 1 ddfuniium f wcumque^ 
^ducdtur dl db extremitdte ^is^ Actefto ^leinde qu$litit 
funHo n ittfeCHotteydffUcetitr nOt^fer o dgdtur qp dc^ 
quidifidns iffi ^ydtom dt^dtiurfdrdlUtd ^d f 1 • Erit idm 
fo dd oqj vt fe dd cz^ fiue^t i£ dd iU nemfentt po ddo 
m > $bfdrdUelds ; ErgoreiidnguU f o m ^ p o q funt ^qudlidi 
qudmobrem reiidngulum f o m nqudle trit quddrdt$ otkptJF 
froftercdil reifttmfigursldtus . .^$d^. 




Licet h$c ver$tmfitin$mnife^i$ne c$ni jf$ltnn JtyfcA^ 
d€fiximHSpqtt$tddm^f$ldhjfefb$Ufd$$t ddrcmmSrdm^ 



Lemma V. 

* Sire&dnguUtm zc^ in eedemexU 

• SensfUno cum tridngulo $rth$g$ni§ 
c b f . c$nuertdtur circa mdnetts idtus 
a d d$tuc dd locum rededt vnde tdfit 
moueri. DicodnnttUtmcircutdremde^ 
ftriftumdtrUngule e b {squdUm ^ 



r. 



5<; 




G JVL A B ; 



ftn* 




148 

/r conoidicmidm ijffirhoiico « fui^s dltitn^fit b$ s 
^MteSfMmJti qmdrtd profomondlium fifidt w eh dd \> f iid 
Jnfld b a ddmidm . Verjnm verb Idtns qudrtd sis ffOfirtUm^o^ 
iinmysifidtvt f b 4^ b e , itdddfld b a ddaUdm » 

Connertdtur figstrd vti di^um efi ^ ^re^idngMlMm zc de^ 
fcribdt cylindrum xuius feUU fer nxem c m > onttUigdSoerque 
fraduUdm<(fertEidm ft> donec oum^x^comicnidsiMh^ & 
€um m\ in i. Mdmfefium efi tridngulHm baf defirikere 
€onum ghf> cuiusdxiseH ah. CoMtfidtut iam pecdori co* 
mum ghf djuidiffdnterdxifldnofer ch^ siueferinm^u^ 
Ro , quod quidemfldrmm ereifum stt ddfigurdmgenitricemco* 
ni ^uemfe ddfUnum g h f • EritqfeStio in cono g h fbyperio^ 
idi EtfrofteredfoUdumquod defiribitur k tridnguh mn g> 
siue e b f , circd axem a d , dqudk trit (ftr lemmdfrimurn) co^ 
midihyferbolicodfr^dicsahyferhalddefcrifto . Hsuusdutem 
cosioidis , siue huius hyferbold Idtus rectum hdbetur (fer lemm. 
Oir4'Ax ^^'^^'J^ifi^^^t nm, ddmg^ita cujsiuedufld bdLdddUdi 
^ . yerfum "uero^ quod efi n iphdbebitury si/ids vt ^mddmt^itd 
txkyjiueduftd ba dddUdmquu erit nu ^od erdtf^c. 

Tbeoremd^ 

Cochlea prlma!rc!uolutionis , qiue Jefcribitur d triangulo e 
if inprascedenti figura, asqualis eft conoidi cuidam hyperbo- 
licOfCUiusaltltudolit ir^ilatusredumfitquartaproportiooa* 
lium»fi£atvt ^i ad bfj ita<dupla ^^ adaiiam. Verfom 

verolatusfitquartaproportionalmmyfifiatvt/l^ ad ^^>ita 
dupla iiiadalianu 

Hoc enim p^ltet ex iam demonft ratis • Praedi^cfitifi cochlea 
af^qualis eft (^per lem. prknum ) amiiillo fa6o atdanguio e bf. 
Sed annulus circularis trianguli ri/praedi^ conoidi eft ai- 
qualis ( per kmma pricedens .) £rgo patet quod propofii 
erat« 



»• 



Scho» 



, rpeCocblea. 149 

Scholtum . 

CocbUd verh cMiMsjtgufdgenitriXfdtdlUlogt^mnmm rcStMm 
y^MlMmJiiyfqudUscB cyUndfocMtMs dtitMdofit e b, cddtmcMm 
^UtittidinefgMrMgeMtrUiSyfemidiMmtter vero hafis mediapro^ 
fertiondisfit inter f b , t^rt04m comfofitMm eie ex f a , a o • 

Si verofigMrdgemtrix eircMUsfMerit ^ eritfe^fd cochUa fri^ 
ma reMolMtionis ad ffharam circMU gemtcrisyVt ferifharia qMa 
defcribitMT 4 radio , ^iMtfit aqMaUs vtrujMe^ nemfc re{if a b m 
fracfdenti fignra^ femidtdmetroqi^ citcttU genitoris ^ oddMas 
ter ias diometri eistfdem circttUgenitoris p 

REliijMMm ejfet vt Mtchdnica etiam Theoremata horMmfi^ 
Udornm neqMtfemMr yfrafertim qttando CochUagigtti^ 
tMratriangMU:CentrMmenimgraMitatisinaxetfiy diMiditq; 
fortinncMlam qnandam iffitts axis (aqttalem atfcittdendam U-^ 
teri e b , c^ circa If^nCfMm mfdittm iffiMs axis colUcamdam) 
veUtti conoidis CMtMfdam hjftrtolici centrttm fecas frofriam 
diametrnm \fiMefradi{lafortiM»tcttlffemiJfem itadiMidit , vti 
tattdemfecaret ctnttMm grastitasis cttittfdamfegmentiffharici 
dttfUtm habtntis aUitMdineyiafitnq^ dato cttitlam circMlo aqttd^ 
. Itm • Sedtanti non eHfingttUs ifias nngas Ungiits frotrabert^ 
ott ie ieneMoUtm LeSForem vUtrins adhttc torqneamMs . Fot^r 
tafif etiamfiet , nifivniMtrfa hacy qnain iBU liieUis contineto 
tMT^ ^ibi difflicttifie comferiam , vt ta qna hic defiderantttr , ^ 
mMltoflura circagraMitaftm , ifSnfqi centrnm yfecMliariUieU 
U Geotnetrice comfrthettdam . Inttrimfcto mt fatrocinitmt dt^ 
itrt Ungijfma tot mtnfinm dtfidia : cMm iamfMfra annttm , tx 
qMo^f^cuU hac ptanimis Gtomttrisfromifsa fMnt^frodMcatMr 
ioftti^md tdrttm itnfreffto . qModqMtdtmflsat^Ms ^e coMfisfa- 
iiMm efi ; neq\ hoc tam negVtgentit meain^MPtttdtnn efi , qMom 
fortMitisqMiiMfdamcafiiMs^infferatifqttc. Acctdit enim in^ 
termedio hoc timfore y vtflMrinm ttttttfittm HMdio atq; labort 

infoUttionem ^tici iWms frobUmdtis tamdiitfa^ 




qMiJsti^€$iitis vldelicetf^wtAeffe debejmffitpirfciei 'uitr^rmm, 
qus ddnjifim Tetefcofy eUbnrdntur . Exitus demenHrMi^nem 
cenftmduit .qtutmqttAm^enim neqtu oftAtAmfigurjtm (vtcre' 
dib,il€ €fi)feifeCiy habefent , nequt "undequJtqke dbfdmtM ytjr 
ferfoUtd i Tirone ddhuc inexfetto , idr inexttcitMo^ider»^ 
iur i4ife^dmen^ (^ vifgurdJlliusddqujmfre^fne tMntnm uc^ 
'€edtbdntydd.eumafq\perfe£iionisffrdd$mftruenerunfy f%rsTe* 
lefcopid oftimi cuiufq; dTtificis^ususddhunediemfamnin hdc 
Vrbe innotturit fup^rduerint . Neque iudicium hocperperim 
proldtum efi ;fed repetitisfepius fiemmdq\cum diUgentia mt- 
rqs experitnentis , no£le , dieque^ (^^dhibitis eruditfjfimis te^ 
Mibusyquorumiudicium^nemoiurzddmndutrit . ^Certe^qudk- 
€unq;fiietit inuentumynefcio plufne gdudq^ldudifq^ mihi dt^ 
iulerit.dnpt^mq xquandoquidem S^enijfimi Mdghi Ducis ejfio^ 
fd , (^vtte H egid UbetdUtds mdgno duti pondere dondtstm mm 
nonfemelHSoluit . Mitum itdq; videti^non debtt quhd omifsi 
pet integtumfemefire UbeUorumjcuri^ tttdmoperdtknomoinue^ 
tOy mihiq; in primis exoptdtsjfiiho yne dicdmi/tilijfimoy impem* 
derim . FdHumetidm efl vt hac decdufd iibells minus cdfii^ 
gati eudferintjduthoreniinirnmdiflrdiioy drdddlidy tdq; eUuer 
fffimd yconuerfo . ^ludpropterwatidus etidm dtq\ eiiam es be^ 
neuole leifor , tit hac qudUatunqvdqui , boniq;fiUrids , e^ etrd- 
tdvel toleres y velcottigds . ptffettim cum tammidnifeftdpU^ 
^runq.fint , v/ neminemfifgete t/Mednt yfedvttrofefe i^ offe^ 
tdnt ^nst videreefi inprin^fidtimrrp^^d:ntmc$epdt9ridy ^fa 
hindefdtisfrequenter in qsquafequitntttr . ^Coinrefiioi^ex non 
^dddMmsinfineifferisyVtpleriq^fole^ vdca^ 

m^ temporis ddmendofd omnid ddnotdttdd . . neq. %^Utimus 
Mutili breuiq^rtcetifionedtiquotdrrdtorumyiomnem ideindeex^ 
\tufdtioni mef locum etripete ; dtim idcitdptftermiffio eorum^ 
qufctnfum^ffi^jfeuty tdmqttdmdpprobdtionisquodddnijgemm 
mihipotitiffetimputdru 



A 



r i^ 1 ^. 



\ \ 



CBe- 



115 

EReu, M. OltIo Manottiveda fe nella prefente Opera R con- 
tenga^cofa che repugni alla Pieta Criftiana» e buonl coilu« 
mi ^c rifcri^a ^ £>• ii di 3 o. di Marzo 1 644: 
Vif$c€i$XioRdbdtii4L VicGef^. di Fir^ 

l^oP.CafolusdeMariottisnultaniin hoc opere contra pie«^ 
tatemac bonos mores inueni labem >immo maximamin ip- 
fo mathematicx ftudentibus» ac huiufcemodi incumbenti* 
busartiinlegendo fum expertus vtilitatem : in quorum fi* 
dcmfcripfi 

ldeinE^$ qkifufrAmAHHprofrid •. 

Attentaprasfemirelationein^matur opus feruatis/eruaflu 
D»die 9*. Aprilis i (544. 

Vimcntins RdbdtuVic.JS€n.Jilori 

Sipuo ftampare in Fiorenza li 1 5. Aprile 1 544.. 

JFr.Idc$moJd CdSiigUone Canc Ml S .OffJemdnd. 

Meffdndro Vtttm Sendtore And.diS.A. Serenifs. 



«« *
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